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near a point solution xi ai (i 1, 1 *, n) when certain conditions are satisfied. In the present paper these conditions are replaced by less stringent ones; the methods of proof of certain existence theorems are very similar to those employed in the first paper, and these proofs are given here in an abbreviated forin. In addition, the asymptotic properties of certain trajectories are discussed by an extension of the methods of Bohl. On account of the more complicated form of certain quadratic forms which occur here, it has been convenient to leave undetermined certain constants which are determined explicitly in the first paper; this procedure, together with the algebraic method of transforming the canonical equations, reduces to a small amount the results common to both papers.

Let F be either a field of char 0 with all roots of unity or a field of char p > 0. Let G be a countable group. Then all simple NIG]-modules are injective if and only if G is locally finite with no elements of order char F and G has an abelian subgroup of finite index. The condition that all simple modules over a ring be injective first appeared in a theorem due to Kaplansky: a commutative ring satisfies the condition if and only if it is von Neumann regular. Several people have studied the property for noncommutative rings, a recent example being [3]. The authors of that paper suggest the problem of characterizing group algebras with this condition. In this paper we make substantial progress by offering Theorem 3. Let F be either a field of char 0 with all roots of unity or a field of char p > 0. Let G be a countable group. Then all simple FtG]-modules are injective if and only if G is locally finite with no elements of order char F and G has an abelian subgroup of finite index. The proof is divided into three parts. In ? 1, we show that F is injective as an 14G]-module if and only if G is locally finite with no elements of order char F. In the second and crucial section, we show that for a certain class of rings ("locally Wedderburn algebras") the condition that all simple modules are injective is equivalent to the property that all simple modules are finite dimensional over their commuting rings. In -?3, we prove the main theorem by showing that if all simple modules are finite dimensional over their commuting rings then G is abelianby-finite. We would like to thank D.S. Passman for suggestions that shortened and improved our work. 1. Results of Villamayor. Most of this section can be gleaned from several of Villamayor's papers. However, the statements and proofs that appear here are new. The field F is a right 14G]-module under the trivial action (i.e. if k E F, then k * a = k(a)e where E is the augmentation map). We characterize those G with F being 14G]-injective. By considering maps from ideals of FtH] to F, the reader can casily prove Received by the editors February 5, 1973. AMS (MOS) subject classifications (1970). Primary 16A26, 20C05; Secondary 16A52, 16A64.

O. Introduction Let (X, Y) be a locally convex linear topological space with topology Y and denote by C its cone of continuous seminorms. Then the two main results of this paper are a bipolar theorem for subsets of C and a Grothendieck-type representation of the completion of C with respect to some topologies of uniform convergence on subsets of X. The completeness theorem is based on an approximation lemma for seminorms corresponding to the one for linear functionals; we prove: If M is a convex circled subset of X and h a seminorm on X whose restriction to M is continuous then for every e > 0 there exists a continuous seminorm h~ on X such that [(h-h~) (m)l < e for all m e M. These results will be established in Sections 1-3 by sublinear techniques in the somewhat more general framework of sublinear functionals. The completeness theorem will provide some new insight in properties of locally convex spaces from the point of view of seminorms and will, consequently, turn out to be a very useful tool in considering special types of those spaces, both in characterizing and handling them. Moreover it will yield a unified approach to several problems in locally convex theory. It is the primary object of Sections 4-6 to carry out some work in this direction. In Section 4 some types of locally convex spaces are characterized by completeness properties of their cones of continuous seminorms. Section 5 is devoted to the problem of characterizing the finest locally convex topology q on X which agrees with the initial topology ~of X on every member Ak of an increasing sequence (Ak)k~ of convex circled subsets of X whose union is absorbing. This problem has been investigated in detail in the papers of Garling [5] and Roelcke [14]. As for this problem the above mentioned Grothendieck-type representation theorem will show that the cone C, of t/-continuous seminorms on X is nothing else but the completion of C with respect to the topology of uniform convergence on the sets Ak, k e N. This result is entirely analogous to the corresponding one concerning the dual space of X with respect to t/(see [14]).

The main new difficulties encountered are the possible existence of inseparable elements in the finite extension ft/P, and also the occurrence of constant field extensions. The former phenomenon is overcome in §1 by the device of replacing P by a suitable purely inseparable extension K s fi for which " descent by norms " to P behaves in a reasonable manner. We then attempt to imitate where possible the processes of [1] applied to Q.JK. The presence of constant field extensions inevitably complicates the analysis, since the " obvious " analogue of Cebotarev's density theorem is unexpectedly false in this case. We append to this paper a section on the appropriate Cebotarev theorem which, we hope, will fill the gap in the literature of this intriguing topic. Section 2 of this paper corresponds broadly to §1 of [1], and in it we prove THEOREM 1IA. Let G¥(q s ) be the constant field ofd, a finite extension of P. Then the number of integral ideals ofk[X] which have degree d, are prime to a given ideal n, and are norms of fractional ideals of Cl is zero unless f divides d, in which case the number is asymptotically

In a recent paper, Pu has shown that every topological space which admits an Abian structure is regular. In this paper we present an example of a regular space which does not admit an Abian Structure to answer a question raised by Pu.

For a domain R, an it-module A is called a Baer module if Ext £ 04, T) = 0 for every torsion /^-module T. Dual to Baer modules, a torsion it-module B is called a UT-moάule if Ext \(X, B) = 0 for every torsion free it-module X. In this paper properties of these two types of modules will be derived and characterizations of Priίfer domains, Dedekind domains and fields will be obtained in terms of Baer and l/T-module properties. One characterization will show the Baer modules are analogous to projective modules in the sense that a domain R is Dedekind if and only if, over R, submodules of Baer modules are Baer. In addition, just as a semisimple ring S can be characterized by the property that all 5-modules are injective, or, equivalently, aU 5-modules are projective, a domain R is a field exactly if every torsion it-module is UT or, equivalently, every torsion free it-module is a Baer module. Further properties of these two kinds of modules will provide sufficient conditions to bound the global dimension of a domain R. 0. Historical Note. The concept of a Baer module goes back to 1936 when R. Baer, in [1], proposed the problem asking for a complete characterization of all abelian groups G such that Extz (G, T) = 0, for all torsion abelian groups T. At that time he showed that any such abelian group must be torsion free, and free if it had countable rank. Then in 1959 R. Nunke, in [10], extended these results to modules over a Dedekind domain, proving that such a module was again torsion free, and projective if it had countable rank. Finally in 1969 P. Griffith, in [5], completely solved Baer's problem for abelian groups, and showed that any such abelian group, now called a Baer group or 2?-grouρ, must be free. In [6], the author extended Griffith's techniques to modules over a Dedekind domain, showing that if A is a Baer module over a Dedekind domain then A is projective. The major adjustments needed in this transition from abelian groups to modules over a Dedekind domain were accomplished by means of Corollary 2 on p. 279 of [12], Theorems 3 and 5 in [8], Lemma 8.3 and Theorem 8.4 in [10], as well as the exposition on ideals and valuations in § 18, 19 of 1. Preliminaries. Unless additional restrictions are stated, in this paper R denotes an arbitrary integral domain: that is, a commutative ring with 1 having no zero divisors. The quotient field oίR will be denoted by Q.

In this paper the methods of M. Karoubi (MR 41 #6205) are generalized to the case of equivariant K-theory. The sets of Fredholm operators in certain (Hilbert) spaces of representations of finite groups G are described which are classifying spaces for equivariant K-functors. The results were announced in the paper MR 46 #2702. Bibliography: 16 items.

In this paper we study the structure of the Banach space K(E, F) of all compact linear operators between two Banach spaces E and F. We study three distinct problems: weak compactness in K(E, F), subspaces isomorphic to l~ and complementation of K(E, F) in L(E, F), the space of bounded linear operators. In § 2 we derive a simple characterization of the weakly compact subsets of K(E, F) using a criterion of Grothendieck. This enables us to study reflexivity and weak sequential convergence. In § 3 a rather different problem is investigated from the same angle. Recent results of Tong [20] indicate that we should consider when K(E, F) may have a subspace isomorphic to l~. Although L(E, F) often has this property (e.g. take E = F =/2) it turns out that K(E, F) can only contain a copy of l~o if it inherits one from either E* or F. In § 4 these results are applied to improve the results obtained by Tong and also to approach the problem investigated by Tong and Wilken [21] of whether K(E, F) can be non-trivially complemented in L(E,F) (see also Thorp [19] and Arterburn and Whitley [2]). It should be pointed out that the general trend of this paper is to indicate that K(E, F) accurately reflects the structure of E and F, in the sense that it has few properties which are not directly inherited from E and F. It is also worth stressing that in general the theorems of the paper do not depend on the approximation property, which is now known to fail in some Banach spaces; the paper is constructed independently of the theory of tensor products. These results were presented at the Gregynog Colloquium in May

This paper is concerned with the Hopf algebra structure of a certain subalgebra S of the Steenrod ring AG of stable cohomology operations in the complex cobordism theory and the sympletic cobordism theory MG*( ), where G=U or Sp, the infinite dimensional unitary or sympletic group, respectively. The Hopf algebra structure and its applications of Steenrod algebra with coefficients Zp, for a prime p, have long been studied by many topologists (Steenrod-Epstein [6]). Novikov [3] investigated the Steenrod rings of generalized cohomology theories. Landweber [1] also studied general properties of AG as Hopf algebra. The main purpose of this paper is to determine the explicit product formula in S (Theorem 3.1) and the indecomposable quotient S/S2 (Theorem 4.1), where S denotes the kernel of the augmentation S-»Z. We use the following notations. Let Z be the ring of integers and Zm = Z/mZ. According to Landweber [1], AG can be expressed as AG = A®S, with the coefficient A = Q^MG*(point). In case G=£7, /t=Z[xl5 x2,...], deg(Xj)=-2/. In case G = Sp,A has not been determined completely. The subalgebra S is a Hopf algebra over Z and has a Z-free basis {Sj}, with dQg(SI) = d^rrir9 where d=2 or 4 according as G=17 or Sp and I=(il9 i2,...) is a sequence of non-negative integers such that all but a finite number of ir are zero. For two Z-graded modules M=^i^L_aoMi and N^^^-^ Ni9 the completed tensor product

then it is fairly well-known that there is an exact sequence for homotopy, homology and cohomology functors which does relate T(g) and T(h) to T( f ) . We regard such an exact sequence both as a special case of the result we aim at and as the axiomatic jumping-off point for the abstract algebraic theory which is developed and applied in this paper. In a previous paper [10], to which this may be regarded as a sequel, we established the machinery of exact couples and spectral sequences in an abelian category 9~. In particular we studied the convergence problem in its fullest generality, and established the exact sequence (Theorem 4.16 of [10])

An epimorphic extension of a ring R is a ring S for which there is a homomorphism 11: R ~ S which is both a monomorphism and an epimorphism in the category of rings. This paper is concerned with epimorphic extensions which are left-fiat in the sense that ~ induces on S the structure of a fiat left R-module; an example of such an extension is the (classical) ring of right quotients of R. Flat epimorphic extensions of commutative rings have been studied by Lazard [-8]; the more general situation in which ~ need not be monomorphic has been considered by Silver [-10]. Characterisations of left-fiat epimorphic extensions of a ring R are given in Section 3, from which it follows that such extensions are rings of right quotients of R in the sense of Utumi [121. Certain properties analogous to those possessed by classical rings of right quotients are also discussed. The principal result of the paper is Theorem 4.1, in which the existence of a left-fiat epimorphic extension P(R) of a ring R in which every other such extension can be uniquely embedded is established. It is shown, in Section 5, that, for a given ring R, P(R) is a semisimple artinian ring if and only if R contains no infinite direct sum of non-zero right ideals and the right singular ideal of R is zero. Finally, commutative semiprime rings are considered. It is shown that, if such a ring R contains all the idempotents of its complete ring Q (R) of quotients, then P(R) is the minimal regular subring of Q(R) which contains R. Examples of such rings are commutative semiprime rings which are integrally closed in Q (R) and commutative Baer rings. It has been drawn to my attention that some of the results of this paper are in the Notes [-3, 41 and [-91. Theorem 4.1 is announced, without proof, in [-91. A stronger result than Corollary 3.4 appears in [31 and [91 and Theorem 3.6 (iii) appears in [41. 2. Notation and Known Results

Introduction. Auslander and Goldman [4] have studied maximal orders over discrete valuation rings. In this paper, relying heavily on their results, we investigate maximal orders over regular local rings of dimension two. The two theories are rather different. We begin ?5 by listing three theorems from [4]. Then we prove a partial generalization of two of them in dimension two (Theorem 5.4) and exhibit two examples which show that the remaining statements do not generalize. In [4] a structure theorem is proved for maximal orders over discrete valuation rings which is the analogue of the Wedderburn structure theorem for simple artin rings. In ?6 we extend this theorem, in a weakened form, to maximal orders over an arbitrary integrally closed noetherian domain R. We sharpen this somewhat when R is regular local of dimension two and the maximal order is well behaved. Various other structure theorems are given in this section. The first four sections are devoted to building up homological machinery, most of which is applied to orders in the last two sections. The setting is fairly general: (R, mn) is a commutative noetherian local ring with maximal ideal m, and A is an R-algebra which is finitely generated as an R-module. The best results (Theorems 1.10 and 2.16) are obtained when A is quasi-local (i.e., A/Rad A is a simple artin ,ring, where Rad A is the Jacobson radical of A). Such rings behave very much like commutative local rings. The author wishes to express his gratitude to Professor Maurice Auslander for his many helpful suggestions and his patient supervision of the research for this paper, which is the major portion of the author's doctoral dissertation. Many thanks also to Professor David Buchsbaum and Dr. Silvio Greco for useful and stimulating conversations. Notations and conventions. Throughout this paper all rings have units and all modules are unitary. We use the abbreviations pd, inj dim, and gl dim for projective, injective, and global dimension, respectively. Only when there is a possible leftright ambiguity will we write l.pd or r.inj dim, etc. R is a commutative noetherian ring and A is an R-algebra which is finitely generated as an R-module. For the first four sections R is local with maximal ideal m. By R& we will mean the completion of R in the m-adic topology, and if M is an

Auslander and Bridger have shown that, under conditions somewhat weaker than finite projective dimension, the "torsion freeness" properties of a module M (e.g. being reflexive, being the kth syzygy of another module) are determined by certain arithmetic conditions on the Exti(M, R). In this paper it is shown that a single ideal, the intersection of the annihilators of these modules, gives this same information. This ideal is then related to the Fitting invariants and invariant factors of M, and a computation is made of certain syzygies of a quotient of M (by a regular M-sequence). Introduction. In [2] it is shown that under certain reasonable hypotheses (somewhat weaker than having finite projective dimension), the torsion freeness properties of a module M are determined by the grades of the modules Exti(M, R) for i>0. These numbers, in turn, depend only on the ideals Wi=rad[Ann Exti(M, R)]. Denoting by y(M) the intersection of these ideals, the author has shown that y(M) likewise determines the torsion freeness properties of M. In this paper this result will be partially generalized, and an explicit calculation made exhibiting the torsion freeness properties (?3). In ?2 we relate y(M) to the Fitting invariants and invariant factors of M. Unless otherwise specified, all rings will be commutative and noetherian and modules finitely generated (hence finitely presented). 1. The ideals ,B(M) and y(M). Although M is projective if and only if Ext1(M, -)=0, one need not test Ext1(M, -) on all modules. Let us write QM=Q1M=ker(P-+M) where P-+M is any map of a projective module P onto M. If we agree to call two objects A and B projectively equivalent when A +P-B+Q for projective objects P and Q, then it is well known that the projective "equivalence class" of QM depends only on that of M (see Lemma 10 below). LEMMA 1. M is projective if and only if Ext1(M, Q2m)=0. Received by the editors February 5, 1971. AMS 1970 subject classifications. Primary 13C10, 13C15, 13DO5.

Introduction.3 Let 2 be a finite-dimensional, simple Lie algebra over an algebraically closed field 5 of characteristic zero. In this paper we shall study the family of all linear irreducible (finiteor infinite-dimensional) representations of S which admit a one-dimensional weight space. It is well known that this family includes all finite-dimensional irreducible representations of 2. 1lore generally, we know that the weight space corresponding to the dominant weight function of Harish-Chandra's irreducible representations with dominant weight functions is one dimensional [5]. Finally in a forthcoming paper, Bouwer studies a class of linear irreducible representations, called standard representations, which do not possess a dominant weight function but do admit a "characteristic weight function" whose weight space is one dimensional. It is the purpose of this paper to provide a characterization of all linear irreducible representations of 2 admitting a one-dimensional weight space.

In a previous paper a method was given by which the efferent activity of an idealized neural net could be calculated from a given afferent pattern. Those results are extended in the present paper. Conditions are given under which nets may be considered equivalent. Rules are given for the reduction or extension of a net to an equivalent net. A procedure is given for constructing a net which has the property of converting each of a given set of afferent activity patterns into its corresponding prescribed efferent activity pattern.

Introduction. The subject matter of this paper belongs to the general theory of sets. The objectives here are the examinations of the structure of classes of sets which are closed under various finite and transfinite set operations and those transformations which preserve these operations. The major results of this paper are partial extensions of (a) the well known theory of a-algebras (e.g. [4]), (b) the work of E. Marczewski [1] on isomorphisms, (c) the work of R. Sikorski [2] on o-homomorphisms and (d) the work of A. Tarski [3] on fields of sets and set functions. Three classes of sets-the algebra, m-algebra and total algebra-as well as four transformations-the m-homomorphism, weak isomorphism, m-isomorphism, and total isomorphism-are studied. A class 3C of sets is an algebra if it is closed under finite set operations, i.e. addition of two sets and complementation. Analogically 3C is an m-algebra if X is closed under m-operations, i.e. complementations and addition of not more than m sets, where m is an arbitrary fixed cardinal number. X is a total algebra if it is closed under all operations, i.e. complementation and arbitrary addition. Two classes of sets, 3C and ?, are weakly isomorphic if 3C and ? considered as partially ordered by the relation of proper inclusion are similar. 3C and ? are m-isomorphic if they have the same properties from the point of view of m-operations; and, finally, X and ? are totally isomorphic if they have the same properties from the point of view of all operations on sets. The most important structure theorems concern the m-operations for mr_n&. The formulation of the corresponding finite and total structure theorems follow easily from the m-theorems and will be omitted except in special cases. The work is divided into two parts: (1) algebras of sets and (2) homomorphisms. In (1) the existence, composition and construction are treated; and, further, the relation between algebras and the "natural set units" is developed. In (2) are discussed some properties of m-additive, complementative transformations. Homomorphisms

A classical question for modules over an integral domain is, "When is the torsion submodule t(A) of a module A a direct summand of AT' A module is said to split when its torsion submodule is a direct summand. Kaplansky has shown [-14] that if R is a Dedekind domain, then every module whose torsion submodule is of bounded order splits. The converse of this result has been shown by Chase [5]. Results of [15] and [3] show that every finitely generated module splits if and only if R is a Priifer domain. Finally, if every R-module splits, then Rotman has shown [19] that R must be a field. Recently, many concepts of torsion have been proposed for modules over arbitrary associative rings with identity. Almost all of these are special cases of "torsion theories" in the sense of Dickson [6]. Moreover, most of these torsion theories are hereditary (i.e., the submodule of a torsion module is torsion); and hereditary torsion classes are classes of negligible modules associated with a topologizing and idempotent filter of left ideals in the sense of Gabriel [12] (also see [17]). Some recent papers ([4], [7], and [11]) have dealt with splitting results for specific hereditary torsion theories over certain commutative rings. The main purpose of this paper is to continue the investigation of the splitting properties of a torsion theory of modules over a commutative ring. Some characterizations for the splitting of modules, whose torsion submodules have bounded order, are obtained (see definition of bounded order below and Theorems 2.2 and 4.6). In particular, these results generalize the abovementioned theorems of Chase [5] and Kaplansky [14]. Our results show that the splitting of modules whose torsion submodules have bounded order frequently forces non-zero torsion modules to have non-zero socles. This increases our interest in the smallest hereditary torsion class 6e containing the simple modules. The class ow has previously been used in the study of commutative Noetherian rings and (left) perfect rings (e.g., see [7] and [10] and their references). For a commutative ring R, we show that 6e(A) is a summand of each R-module A if and only if non-zero R-modules have non-zero socles. This generalizes the main results of [7] and [11]; moreover, in case R is a Dedekind domain, our result coincides with the above result of Rotman. We also examine the properties of a splitting hereditary torsion theory of modules over a local ring R (unique maximal left ideal). We show that if R possesses a non-trivial torsion theory (~--, #-) such that every finitely generated module splits, then R is an integral domain, and (Y-, ~ is Goldie's torsion

In this paper the group of homeomorphisms of an arbitrary topological manifold is considered, with either the compact-open, uniform (relative to a fixed metric), or majorant topology. In the latter topology, a basis of neighborhoods of the identity is given by the strictly positive functions on the manifold, a homeomorphism being in the neighborhood determined by such a function if it moves each point less than the value of this function at the point. The main result of the paper is the proof of the local contractibility of the group of homeomorphisms in the majorant topology. Examples are easily constructed to show that this assertion is false for the other two topologies for open manifolds. In the case of a compact manifold the three topologies coincide. In conclusion a number of corollaries are given; for example, if a homeomorphism of a manifold can be approximated by stable homeomorphisms then it is itself stable.

Let f be a real-valued function defined and finite on sets from a family F \mathcal {F} of bounded measurable subsets of Euclidean n-space such that if T ∈ F T \in \mathcal {F} , the measure of T is equal to the measure of the closure of T. An earlier paper [Trans. Amer. Math. Soc. 145 (1969), 439-454] considered the questions of finiteness and boundedness of the upper and lower regular derivates of f and of the existence of a unique finite derivative. The present paper is an extension of the earlier paper and considers the summability of the derivates. Necessary and sufficient conditions are given for each of the upper and lower derivates to be summable on a measurable set of finite measure. A characterization of the integral of the upper derivate is given in terms of the sums of the values of the function over finite collections of mutually disjoint sets from the family.

This paper contains a complex version of Theorem 5.4 (Dominated Extension Theorem) from the paper “Structure in real Banach spaces” by E. Effros and the present author (to appear in Annals of Mathematics).

that this question is intimately connected with the existence of certain kinds of simple radical rings; it is this connection which we wish to emphasize in this paper. We define a left valuation ring to be a nonzero ring R in which, for every x, y in R, there exists a u in R such that either x = uy or y = ux. A right valuation ring is defined similarly. (A valuation ring in the sense of [1] above is a left and right valuation ring, in the sense of this paper, without divisors of zero.) There do exist left valuation rings which are not right valuation rings. An example can be given modifying [2, ?6, p. 219]. The construction is, however, omitted. We now prove some results valid for left valuation rings to lead up to the connection with simple radical rings. The lesser ones are called simply results, the more important ones theorems. RESULT 1. The left ideals of R are linearly ordered by inclusion.

The theory of adjoint functors has been used by Morita to develop a theory of Frobenius and quasi-Frobenius extensions subsuming the work of Kasch, Miller, Nakayama, and others. We use adjoint functors to define a pairing of the two rings and develop a theory of relative projective and injective modules for pairings generalizing that of Hochschild for extensions. The main purpose of this paper is to define "strongly separable pairings" generalizing strongly separable (i.e. finitely generated projective separable) algebras. We show that such pairings have very close connections to category equivalences, so that it is natural to investigate those properties shared by two rings which admit a strongly separable pairing. We show that most "categorical" properties are so shared. Introduction. Throughout this paper we shall assume that all rings are associative and have an identity, and that all modules, subrings, and extensions are unital. We shall further assume that all functors between categories of modules are covariant and additive. Frequently in ring theory we find two rings, R and S, that are related in some specific way. From our knowledge of this relation we wish to study the structure of one of the rings in terms of the other. Familiar examples include (1) algebras over a commutative ring, (2) general extensions of a given ring, (3) endomorphism rings of finitely generated projective modules. In each of these examples the relation determines what we shall call an " adjoint triple" of functors between the categories RX/ and s, of left modules over R and S. As Morita observed in [20], much of the relative structure of R and S is determined by this adjoint triple. Let R and S be rings. For covariant additive functors F: R'1 -? s,# and G: s# RX#, we say that G is a left adjoint of F and that F is a right adjoint of G in case there is a natural isomorphism HomR (G(N), M) Homs (N, F(M)) Received by the editors September 2, 1969. AMS Subject Classifications. Primary 1690, 1720.

Classical Galois cohomological results for purely inseparable field extensions of exponent one are generalized here to commutative rings of prime characteristic. Given a commutative ring extension C over A of prime characteristic p, there are three variants for the Brauer group B(C/A) of central separable A-algebras split by C: the Amitsur cohomology group H2(C/A, Gm), the Chase-Rosenberg group PV(C/A), and Hochschild's group 4(C, g) of regular restricted Lie algebra extensions of C by the Lie algebra g of all A-derivations on C. In this paper we show that if C is finitely generated projective as an A-module and C [g] = EndA (C), then H2(C/A, Gm) , C) ,l(C/A). As a corollary we show that Hi(C/A, Gm) is zero for all i > 2. When C is a field, these are the results of Berkson, Hochschild and Rosenberg and Zelinsky [4], [11], [12]. As in [11] we show that the Lie algebra extensions which arise from central separable algebras are trivial extensions when regarded as ordinary extensions so that the essential structural elements are here precisely those which differentiate the restricted extensions from the ordinary ones. We also show that if R is a commutative C-algebra which is finitely generated, projective as a C-module, then the Brauer group B(R/A) is mapped onto the Brauer group B(R/C). The last result is also due to Hochschild when C is a field [10]. ?1 contains the background on projective modules which came into the picture. Due to their peculiar behavior all relevant automorphisms turn out to be inner which explains why instead of some exact sequences we get two isomorphism theorems. In ?2 the isomorphism of e(g, C) with 91(C/A) is proved. ?3 and ?4 provide the preliminary materials for ?5. ?3 contains an exposition on the theory of differentials in rings of prime characteristic. Its application to Amitsur cohomology is given in ?4. The main results are given in ?5. Throughout this paper C over A always denotes a commutative ring extension of prime characteristic p such that C is finitely generated projective as an A-module Received by the editors October 30, 1969. AMS 1970 subject classifications. Primary 13A20.

The main purpose of this paper is to show that a regular semigroup S is a semilattice of bisimple semigroups if and only if it is a band of bisimple semigroups and that this holds if and only if O) is a congruence on S. It is also shown that a quasiregular semigroup S which is a rectangular band of bisimple semigroups is itself bisimple. In [3, Theorem 4.4] it was shown that a semigroup S is a semilattice of simple semigroups if and only if it is a union of simple semigroups. The purpose of this paper is to obtain corresponding results for a semigroup which is a semilattice of bisimple regular semigroups. Unfortunately, a semilattice of bisimple semigroups need not be a union of bisimple semigroups as illustrated by a simple co-semigroup constructed by Munn [5]. However, we get some equivalent conditions for such semigroups. In particular we show that a regular semigroup is a semilattice of bisimple semigroups if and only if it is a band of bisimple semigroups. 1. Equivalent conditions. In this section we consider a set of equivalent conditions for a semigroup S to be a semilattice of bisimple semigroups. We adopt the terminology and notation of [2]. LEMMA 1.1. Let S be a semilattice Q of semigroups Sat and let D be a D-class of S. Then, either S,()D = Ol or DCS,. PROOF. Suppose S,COD$ EZ. Let aa=SaC,D. If b#ESp and aADbf, then there exists c,CS, ('yCQ) such that aG(RcT and cy2bg. Also a,(Rc,y implies that either a,,=c7 in which case y==a, or there exist x ESx, y,,-S,, (X, ,u in Q) such that a,xx =cy and czyy=a,. However, since S is a semilattice of the semigroups Sa,, axx CS,, and c7y,, ?EST . It follows that y = aX and a =,yg and so y a and a c'y. Thus, in either case, ,y =a. Likewise, y =,B. Therefore, a =3 and DCSSa. LEMMA 1.2. Let S be a semigroup. If ab Dba for all a, b in S, then D is a congruence on S. PROOF. Let ajDb. Then there exists xeS such that aCx and x(6b. Since ? is a right congruence and (R is a left congruence, we have Received by the editors July 3, 1970. AMS 1969 subject classifications. Primary 2092, 2093.

Introduction. In this paper several properties of absolutely pure modules are given. It is shown that absolutely pure and injective are equivalent properties for modules over Dedekind rings. However, it is proved that absolutely pure and injective are not equivalent properties for modules over rings which are not Noetherian. That every module has a maximal absolutely pure submodule is also established. A sufficient condition for the uniqueness of a maximal absolutely pure submodule is also given. This paper constitutes a portion of the author's doctoral dissertation written at the University of South Carolina where he held a Cooperative Graduate Fellowship. The writer is indebted to Professor Edgar Enochs who suggested this topic and directed its development while providing sufficient inspiration and assistance and, most of all, exhibiting infinite patience. In this paper all rings will have a unit and all modules will be unitary. A will always denote a ring. We agree that if E' is a submodule of E and v. E'—>E is the canonical injection then the map 1 ®v: F®E' —>P £ will be called the canonical map where 1: F-^F is the identity map of P. If the canonical map is an injection for all P, then E' is said to be a pure submodule of P. Observe that if E' is a pure submodule of E then aEC\E' =aE' for all nonzero aEA by examining the diagram

where the A's and u are complex-valued functions, such that the above expression has a meaning on a bounded connected region S in the real XYplane. We assume that the A's are bounded measurable functions on S. We shall restrict u in such a manner that (A) may be considered a linear transformation between two Hilbert spaces. In the first five sections of this paper, we study the linear transformations between two abstract Hilbert spaces 11 and 1 2,which are regarded as coinciding only in special cases. While our investigations are naturally based on the modern workt on the subject, they are particularly closely allied to a paper of J. von Neumann.t Roman numerals indicate essentially new results. In ?1 and ?2, we adapt the treatment of (N) to our problem to obtain the elementary theory of such transformations. In ?2, we also discuss the significance in terms of groups of the adjoint. ?3 deals with continuous linear transformations and while most of these results are well known,11 Theorem I is new and is used later. It also can be considered as indicating the "graphical interpretation" of certain general results of J. von Neumann, which are cited in connection with the theorem. Theorem II deals with the solution of

In an earlier paper (3), a non-abelian cohomology theory (in the dimensions 0 and 1) for associative algebras was developed. One of the objectives was to obtain equivalence classes of crossed homomorphisms by considering inner automorphisms of the coefficient-algebra. This paper is an adaptation of the methods employed there to the case of Lie algebras. Throughout, all our Lie algebras will be over the field of real numbers, and finite-dimensional.

This paper considers the Banach algebra with the usual norm and convolution as multiplication. A characterization is given for closed ideals of which are rotation invariant and have as spectrum, in terms of annihilators of certain collections of pseudomeasures. The main result of the paper is connected with a construction which yields an uncountable chain of closed ideals intermediate between neighboring invariant closed ideals with spectrum . This construction associates an ideal with a closed subset . It is shown that if then . Another result is the lack of a continuous projection from the largest to the smallest ideal when , and when , from an invariant ideal onto the neighboring smaller invariant ideal. A certain algebra of functions on the sphere which arises naturally in the construction of the intermediate ideals is also studied.Bibliography: 18 items.

which the functor Tory (N, ): RM -- Ab is naturally equivalent to G. In this paper, we show that if R is a noetherian ring of global dimension 1 (for example, if R is a Dedekind domain, or a ring of triangular matrices over a field), then the class of Tor-functors is characterized as that class of functors G forwhich G is half exact, G preserves direct limits, and G(R) = 0. In [2], Auslander shows that if R is noetherian and F: RM-- Ab is a coherent half exact functor which preserves direct limits, then there is a right R-module N and an exact sequence of functors TorR (N, ) F(R) OR-- F->- TorR (N, ) 0. In ?1, we recall the definition of coherent functors and some of their properties from [2]. We also show that every half exact functor is a filtered limit of half exact coherent functors ("filtered limit" being our term for a generalization of direct limits defined in [1]). In ?2, we show that if R is noetherian and G: RM ->- Ab is a half exact functor which preserves direct limits, then there is an exact sequence for G similar to the one above, provided that a filtered limit of Tor functors is a Tor functor. (If R is of global dimension 1, this condition is satisfied, and our characterization of Tor functor follows immediately.) In ?3, we extend Auslander's exact sequence to the class of coherent rings, namely rings for which the dual of projective modules are flat. In ?4, we let R be a coherent ring and G: RM->- Ab be a half exact functor which preserves direct limits, and obtain a result similar to that of ?2. We also compare the coherent case with the noetherian case. The results in this paper are from the author's thesis done at Brandeis University under Professor Maurice Auslander, whom the author wishes to thank for his encouragement, suggestions, and inspiration.

This paper treats extension and retraction properties in the category *$/9 of compact metric spaces with periodic maps of a prime period p; the subspaces and maps in J^p are called equivariant subspaces and maps, respectively. The motivation of the paper is the following question: Let E be a Euclidean space and α: E X E-> E X E be the involution (x, y) -> (y, x), i.e., the symmetry with respect to the diagonal. Suppose that Z is a symmetric (i.e., equivariant) closed subset of ExE which is an absolute retract; that is, Z is a retract of E X E. When does there exist a symmetric (i.e., equivariant) retraction Ex E-+ZΊ This is an extension problem in the category J2/'p. If X and Y are spaces in J£fp, A is a closed equivariant subspace of X and /: A -> Y is an equivariant map, then the existence of an extension of / does not, in general, imply the existence of an equivariant extension. It is shown, however, that if A contains all the fixed points of the periodic map and dim(X— A) < oo, then a condition for the existence of an extension is also sufficient for the existence of an equivariant extension. In particular, it follows that a finite dimensional space X in Sf 'p is an equivariant ANR (i.e., an absolute neighborhood retract in the category Sf v) if and only if it is an ANR and the fixed point set of the periodic map on X is an ANR. Generally speaking, the paper deals with the question of symmetry in extension and retraction problems.

In the article t A ne?w method of determining the differential parameters and invariants of qutadraitic differential quantics I have shown that the application of a certain symbolic method leads very readily to the formation of expressions remaining invariant with respect to the transformation of quadratic differential quantics. The presentation in that article was only a preliminary one and the work practically confined to the case of two independent variables. In my paper 4 Invariants and covariants of quadratic differential quantics qf n variables a more complete treatment was intended and the investigation applied throughout to the case of n variables, leaving aside, however, simultaneous invariant forms of more than one quantic. The present paper contains in ?? 1-6 and ? 8 essentially the content of the last mentioned paper; the greater parts of ? 5 and ? 8, and all the remaining articles are new, in particular the extensive use of covariantive differentiation.

In this paper a solution of a problem of G. G~TZEI~ [1] (Problem 17) is given, To formulate the problem let 9.1 be a universal algebra (briefly: algebra) and let ~2 (9. 0, 932 (92), 5) (92) be the semigroups of the endomorphisms, monomorphisms, and epimorphisms of 9J, respectively. The problem is to characterize the triplets (~(~), 932(~I), ~(gj)) in terms of the theory of semigroups. G. GR~TZER gives in [1] certain conditions necessary for a given triplet (E, M, H) of semigroups to be isomorphic with some triplet (~(~.I), 9Jl(92), ~)(~l)) (see w 1, Theorem, conditions C1--C3, in this paper). E. FR~D and (later) the author found a further necessary condition, independent of the former ones (condition C4). Conditions C1--C4 are jointly sufficient; consequently, these give the solution of the problem. In w 1 we fix some necessary notions, formulate the theorem, prove the necessitystatement of it and outline the construction, serving to prove the sufficiencystatement. In w 2 we describe the latter construction in detail and give, as corollaries of the theorem, solutions of Problem 16 in [1] concerning the characterization of the couples (@(9.I), 93l(92)) and the corresponding problem for epimorphisms. I must express my gratitude to MR. GEORGE GR~TZER for his valuable remarks in connection with this paper.

This paper centers around the theorem that a commutative ring R R is noetherian if every R P , P {R_P},P prime, is noetherian and every finitely generated ideal of R R has only finitely many weak-Bourbaki associated primes. A more precise local version of this theorem is also given, and examples are presented to show that the assumption on the weak-Bourbaki primes cannot be deleted nor replaced by the assumption that Spec ( R ) (R) is noetherian. Moreover, an alternative statement of the theorem using a variation of the weak-Bourbaki associated primes is investigated. The proof of the theorem involves a knowledge of the behavior of associated primes of an ideal under quotient ring extension, and the paper concludes with some remarks on this behavior in the more general setting of flat ring extensions.

In the first part of this paper we prove the following extension theorem. Let P q ∗ P_q^ \ast be a q q -dimensional punctured polycylinder, i.e. a product of disks and punctured disks. Let W n {W_n} be a compact complex manifold such that the bundle of holomorphic q q -forms is positive in the sense of Grauert. Let f : P q ∗ → W n f:P_q^ \ast \to {W_n} be a holomorphic map whose Jacobian determinant does not vanish identically. Then f f extends as a rational map to the full polycylinder P q {P_q} . In the second half of the paper we prove the following generalization of the little Picard theorem to several complex variables: Let V ⊂ P n V \subset {P_n} be a hypersurface of degree d ≧ n + 3 d \geqq n + 3 whose singularities are locally normal crossings. Then any holomorphic map f : C n → P n − V f:{C^n} \to {P_n} - V has identically vanishing Jacobian determinant.

Throughout this paper R will be an integral domain, not a field ; Q will be the quotient field of R; and K will be the i?-module QjR. R will be said to have property D, if every torsion-free i?-module of finite rank is a direct sum of /?-modules of rank one. In [9, §6] I attempted to give necessary and sufficient conditions for an integral domain to have property D. I have since discovered, however, that Lemma 6.4 of that paper is false, with the result that Theorem 6.1 is also false as stated, and needs some rather strong additional hypotheses to be resurrected. Other false lemmas based on Lemma 6.4 are Proposition 6.5 and Corollary 6.7; and Lemma 6.8 remains without a valid proof. Compounding the confusion, I have also discovered an error in Kaplansky's proof of his theorem [4] which purported to handle the Noetherian case, and this theorem remains without a valid proof. Proposition 6.11 and Corollary 6.12 of my paper [9] were based on Kaplansky's proof, and are false as stated. Corollary 6.13 remains without a valid proof. The purpose of this paper is to attempt to restore some order out of this chaos. In Theorem 4 I have settled the Noetherian case. This theorem states that a Noetherian domain has property D if and only if it is one of two types of Noetherian domains. However, I have not been able to find an example of a ring of type (1); i.e. a principal ideal domain R with exactly two nonzero prime ideals Px, P2 and such that RPl, RP2 are complete discrete valuation rings. It was precisely this type of ring that Kaplansky thought that he had proved could not exist. Thus one of the remaining unsolved tasks in finding all of the kinds of integral domains that have property D is either to produce an example of this type of ring, or to prove that it cannot exist. In Theorem 3 I have shown that if R is an /z-local domain with more than one maximal ideal, then R has property D if and only if R has exactly two maximal ideals Mx, M2 and RMl, RM2 are maximal valuation rings. The question of whether such a ring can exist is, of course, a generalization of the previous question raised. Theorem 2 proves that a valuation ring has property D if and only if it is a maximal valuation ring. The major part of this result, namely that a maximal valuation ring has property D, is originally due to Kaplansky [3, Theorem 12] who generalized Priifer's result [11] for complete discrete valuation rings. Both of

In this paper we use a method of an earlier paper in order to prove the existence of some symmetric systems of minimal surfaces bounded by curves having self-intersections.

In this paper it is shown that if X is a torus-like continuum, then X has the shape of a compact connected abelian topological group. Let HI be a collection of compact connected Lie groups. In light of the above result it is natural to ask if a H-like continuum has the shape of a compact connected topological group. An example is given to show that this is not the case. Introduction. Let C denote the category of compact Hausdorff spaces and continuous maps and let H: C--HC be the homotopy functor. Let S: C--SC be the functor of shape in the sense of Holsztynski for the projection functor H [5]. It is assumed that the reader is familiar with the equivalence of this approach to shape with that of Mardesic and Segal using ANR-systems [12]. A precise statement of this equivalence with a proof is given in the Appendix of [6]. In this paper we give the shape classification of all torus-like continua. If X is a torus-like continuum, then it has the same shape as a compact connected abelian topological group. It is shown, in fact, that X has the same shape as char H1(X) where H"'(X) is n-dimensional Cech cohomology over the integers. Using our knowledge of the shape properties of compact connected abelian topological groups contained in [6], [7], and [8] several properties of torus-like continua are derived. In light of the above result about torus-like continua, it is natural to ask if a fl-like continuum might not have the shape of a compact connected topological group where rl is a collection of compact connected Lie groups. An example is given to show that this is not the case. In this section of the paper we also show that if G is a compact connected topological group with H'(G)=O, then Hn(G)/Tor Hn(G) has property L for all n_O. This result is invoked to show that the above example cannot have the shape of a compact connected topological group. An example is Received by the editors November 20, 1972. AMS (MOS) subject classifications (1970). Primary 55D99; Secondary 22B99.

A hypergroup is a locally compact space on which the space of finite regular Borel measures has a convolution structure preserving the probability measures. This paper deals only with commutative hypergroups.-§1 contains definitions, a discussion of invariant measures, and a characterization of idempotent probability measures. §2 deals with the characters of a hypergroup. §3 is about hypergroups, which have generalized translation operators (in the sense of Levitan), and subhypergroups of such. In this case the set of characters provides much information. Finally §4 discusses examples, such as the space of conjugacy classes of a compact group, certain compact homogeneous spaces, ultraspherical series, and finite hypergroups. A hypergroup is a locally compact space on which the space of finite regular Borel measures has a convolution structure preserving the probability measures. Such a structure can arise in several ways in harmonic analysis. Two major examples are furnished by the space of conjugacy classes of a compact nonabelian group, and by the two-sided cosets of certain nonnormal closed subgroups of a compact group. Another example is given by series of Jacobi polynomials. The class of hypergroups includes the class of locally compact topological semigroups. In this paper we will show that many well-known group theorems extend to the commutative hypergroup case. In §1 we discuss some basic structure and determine the idempotent probability measures. In §2 we present some elementary theory of characters of a hypergroup. In §3 we look at a restricted class of hypergroups, namely those on which there is a generalized translation in the sense of Levitan ([11], or see [12, p. 427]). (The notation of the present paper would seem to have two advantages over Levitan's: ours is compatible with current notation for compact groups, and in Levitan's notation, it is almost impossible to express correctly commutativity and associativity.) The theory for these hypergroups looks much like locally compact abelian group theory, yet covers a much wider range of examples. Finally in §4 we discuss some examples and further questions. 1. Basic properties. We recall some standard notation (in the following, X is a locally compact Hausdorff space): Received by the editors January 13, 1972. AMS (MOS) subject classifications (1970) Primary 22A20, 22A99, 43A10; Secondary 33A65, 42A60.

Free cyclic actions on a closed oriented (weakly almost complex, respectively) manifold which preserve the orientation (weakly comnplex structure) are considered from the viewpoint of equivariant bordism theory. The author gives an explicit presentation of the oriented bordism module structure and multiplicative structure of all orientation preserving (and reversing) free involutions. The odd period and weakly complex cases are also determined with the aid of the notion of formal group laws. These results are applied to a nonexistence problem for certain equivariant maps. Introduction. As the oriented analogue of the free equivariant unoriented bordism theory Rt*(X, A, r) of Stong [121, K. Komiya and C. M. Wu have respectively defined the free equivariant oriented bordism theories Q4+(X, A, r) and Q*(X, A, r) for involutions (X, A, r) (Komiya [9]), and Q*(X, A, r) for maps of odd prime period (X, A, r) (Wu [17]). The main object of the present paper is to apply Komiya's theories to the geometrical determination of the oriented bordism algebras Q* (Z2) of all orientationpreserving free differentiable involutions and Q(Z2) of all orientation-reversing free differentiable involutions. (Compare with the semi-geometric methods in Stong [11, Chapter VIII]). We also remark in this paper that the equivariant oriented and weakly complex theories of Wu, together with Mis'cenko's theorem [10, Appendix 1], give rise to a simple proof of the structure theorem for Q*(Z p) [2], U*(ZM, U*(Zm) ([3], [5k[6], [7]) and K'NLn(m)) [8]. These results are applied to the nonexistence problem for equivariant maps. In ?1, we define the bordism groups Q4(X, r) and Q(X, r), and then introduce the external product and the Pontrjagin product in these theories. In ??2 and 3, we give two kinds of direct sum decompositions of Q*(S , a) and Q(Sn, a) into isomorphic copies of Q(S, a). Deviating slightly from the Received by the editors July 28, 1971 and, in revised form, February 4, 1972. AMS (MOS) subject classifications (1970). Primary 57D75, 57D85, 57D90; Secondary 55B20, 55C35, 55G37.

Introduction. Let F be a field of characteristic p, and let C be a finite algebraic extension field of F such that CPCF. It has been shown in [4] that there is a one to one correspondence (up to isomorphisms) between the simple finite dimensional algebras with center F and containing C as a maximal commutative subring and the regular restricted Lie algebra extensions of C by the derivation algebra of C/F. This led to a very simple description of the group of similarity classes of these algebras. The main task of the present paper is to investigate the connection between restricted Lie algebras and simple associative algebras of characteristic p quite generally. For this purpose, we generalize the notion of a regular extension of C by the derivation algebra of C/F to that of a regular extension of a simple algebra A with center C by the derivation algebra of C/F. The correspondence mentioned above is then generalized to a one to one correspondence between these extensions and the simple algebras with center F which contain A as the commutator algebra of C (Theorem 3). We shall then show how every simple algebra containing a purely inseparable extension field of its center as a maximal commutative subring can be built up in a series of steps from regular Lie algebra extensions. With a suitable composition of regular extensions, which we define in ?2, this reduces the structure of the group of algebra classes with a fixed purely inseparable splitting field to the structural elements of restricted Lie algebras, at least in principle. However, it does not yield a direct description of this group, as was the case for splitting fields of exponent one. In ?3 we show that the tensor multiplication with a purely inseparable extension field C of F maps the Brauer group of algebra classes over F onto the Brauer group over C (Theorem 5). Although this result seems not to have been stated before, it may well be regarded as one of the culminating points of Albert's theory of p-algebras. In fact, Chapter VII of [1] contains all the essential points of a proof along classical lines, which is the first proof we give in ?3. Here, the main tools are Galois theory and the theory of cyclic p-extensions. Our second proof is entirely different. It proceeds within the framework of the first part of this paper, replacing the field theory with the theory of Lie algebras. In particular, we use the theory of restricted Lie algebra kernels, [3], in order to show that if A is a simple algebra with center C and CPC F then there always exists a regular extension of A by the derivation algebra of C/F, whence (by Theorem 3) A can be imbedded as the

The first theorenm of this paper is an extension of a splitting theorem for finite groups (cf. [61]) to include periodic groups certain of whose Sylow subgroups contain no elements of infinite height. The second theorem is an extension of the principal ideal theorem for finitely generated groups (cf. [71) to include all groups except again that some of the Sylow subgroups contain no elements of infinite height. A counter example will show that this latter restriction is necessary for both theorems. In the first draft of the paper the second theorem was stated for periodic groups, the proof being based onl the first theorem. The present more general statement anld proof independent of Theorem-i 1 are due to the referee to whom I am also indebted for a simplification in the proof of the lemma below.

A “natural system” consists of a Hausdorff space Σ \Sigma plus an algebra A \mathfrak {A} of complex-valued continuous functions on Σ \Sigma (which contains the constants and determines the topology in Σ \Sigma ) such that every continuous homomorphism of A \mathfrak {A} onto C {\mathbf {C}} is given by an evaluation at a point of Σ \Sigma (compact-open topology in A \mathfrak {A} ). The prototype of a natural system is [ C n , P ] [{{\mathbf {C}}^n},\mathfrak {P}] , where P \mathfrak {P} is the algebra of polynomials on C n {{\mathbf {C}}^n} . In earlier papers (Pacific J. Math. 18 and Canad. J. Math. 20), the author studied A \mathfrak {A} -holomorphic functions, which are generalizations of ordinary holomorphic functions in C n {{\mathbf {C}}^n} , and associated concepts of A \mathfrak {A} -analytic variety and A \mathfrak {A} -holomorphic convexity in Σ \Sigma . In the present paper, a class of extended real-valued functions, called A \mathfrak {A} -subharmonic functions, is introduced which generalizes the ordinary plurisubharmonic functions in C n {{\mathbf {C}}^n} . These functions enjoy many of the properties associated with plurisubharmonic functions. Furthermore, in terms of the A \mathfrak {A} -subharmonic functions, a number of convexity properties of C n {{\mathbf {C}}^n} associated with plurisubharmonic functions can be generalized. For example, if G G is an open A \mathfrak {A} -holomorphically convex subset of Σ \Sigma and K K is a compact subset of G G , then the convex hull of K K with respect to the continuous A \mathfrak {A} -subharmonic functions on G G is equal to its hull with respect to the A \mathfrak {A} -holomorphic functions on G G .

1. The Geometry of Numbers was founded by Minkowski in order to attack certain arithmetical problems, and is normally concerned with lattices over the rational integers. Minkowski himself, however, also treated a special problem over complex quadratic number fields [5], and a number of writers have since followed him. They were largely concerned with those fields which have class-number h= 1; and this simplification removes many of the characteristic features of the more general case. Hermann Weyl [10] gave a thorough account of the extension of Minkowski's theory of the reduction of quadratic forms to "gauge functions" over general algebraic number fields and quaternion algebras, and we shall follow part of his developments, though our definition of a lattice is quite different. The desirability of extending the Geometry of Numbers to general algebraic number fields was emphasized by Mahler in a seminar at Princeton. In this paper we shall carry out this program, extending the fundamental results of Mahler [4] to our more general case and applying them to specific problems. Certain new ideas are necessary, but much of this paper must be regarded as expository. In particular, when the proof of a result is essentially analogous to that for the real case we have merely given a reference to the latter. 2. Let K be an algebraic extension of the rationals of degree m. We regard K as an algebra over the rationals, which we can extend to an algebra K* over the reals. It is well known that K* is commutative and semi-simple (being in fact isomorphic to the direct sum of r copies of the reals and s copies of the complex numbers, where r and 2s are the number of real and complex conjugates of K); and the integers of K* are just those of K. We now define the n-dimensional space Kn over K as being the set of ordered n-tuples of elements in K*. Any tCK* is of the form t=x1i1+ * * * +xwmU, where the x, are real and co,, * * *, com is an integral basis for K; and hence there is a natural map of Kn onto Rmn in which each component t is mapped onto m of the components of the point in Rmn, namely xi, x, m as above. We can define a metric and a measure in Kn by means of those in Rmn, with the above map, and so Kn is a locally compact complete metric space.

Introduction. Among the valuation ideals in a polynomial ring = K [x, y] in two indeterminates, the ones of central importance are the simple valuation ideals, that is, the valuation ideals which are not products of two ideals different from 0, since every valuation ideal has a unique factorization into simple valuation ideals. The problem of the characterization of simple valuation ideals has been dealt with by Zariski, in the case that the field K is algebraically closed and of characteristic 0, in his paper, Polynomial ideals defined by infinitely near base points. There the problem is referred to the ring of holomorphic functions in x, y: *= K { x, y }, and a valuation ideal q in Z* is simple if and only if its general element is absolutely irreducible. If, however, only the characterization of the simple valuation ideals is desired, the notions of the general element of an ideal in D* and its absolute irreducibility are somewhat too strong for the problem set; although it should be stated that these notions are applied by Zariski to other topics not touched upon here. In this paper we treat the theory of simple valuation ideals by a more explicit and direct method and we also extend the theory to algebraically closed fields of arbitrary characteristic p 5 O. We characterize the simple v-ideals q in the sequence of zero-dimensional valuation ideals in ?, for a given valuation v, in terms of the value v(q) (under v) of q (that is, the least value assumed by elements of q) and of the least value greater than v(q) assumed by elements of ?). If q is not simple, an explicit factorization for q in terms of the two mentioned values is given Since in our treatment the field K is of arbitrary characteristic, Puiseux series expansions for valuations are not available. A corresponding tool is found in Theorem 6. There for a given valuation v, a certain (finite or infinite) sequence of polynomialsfi(x, y) is introduced. In the case that K is of characteristic 0, if v is given('), for example, by y=clxr(l)+c2xr(2)+ * * * , ciCK, r(i) =ri rational, with 0

A theorem on uniform asymptotic stability of the null solution of a system of differential equations is proved while assuming that the null solution of a limiting equation is uniformly asymptotically stable. This generalizes some of the results of L. Markus. 1. The concept of the set of limiting equations of a given differential equation has been introduced by G. R. Sell [4]. The notion of asymptotically autonomous differential equations introduced by L. Markus [2] can be described as those differential equations for which the set of limiting equations consists of a single point. In [4], G. R. Sell has proved a theorem on asymptotic stability of the null solution of a given differential equation while assuming that the null solution of the given differential equation is uniformly stable and the null solution of every limiting equation is asymptotically stable (in a uniform sense). However, as pointed out in a remark ([4, p. 273] and [5, p. 536]), his theorem does not generalize a result of L. Markus [2, Theorem 2]. The aim of this paper is to prove a theorem on uniform asymptotic stability which generalizes the result of L. Markus [2, Theorem 2]. 2. Throughout this paper, we follow the same notation as in [3] and [4]. Let W be an open set in Rn, Euclidean n-space. The Euclidean norm on Rn will be denoted by Ixi. Let C= C( Wx R, Rn) denote the set of all continuous functionsf defined on Wx R with values in Rn. We shall say that a functionf is admissible [3] if (i)f E C, and (ii) the solutions of the differential equation x'=f(x, t) are unique. By the second condition we mean that given any point (xo, to) in Wx R, there is precisely one solution b of x'=fJ(x, t) that satisfies 0(to)=xo. It is evident that iff is an admissible function, then every translatef, off (where f,(x, t)=f (x, t+i-)) is an admissible function. Also, iff satisfies the global existence property, then so does eachf,. Let F={f,: e c R} be the space of translates off; then F is a Received by the editors September 25, 1972 and, in revised form, November 7, 1972 and January 2, 1973. AMS (MOS) subject classifications (1970). Primary 34C35.