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Introduction. In a recent paper [2] R. H. Bruck has introduced the concept of right neoring and discussed some properties of these systems. In particular, he has considered analogues of certain properties of the ring of integers. This paper is essentially a commentary on Bruck's paper and we generalize some of his results as follows. The construction of the universal right neoring in [2] is applied to the free monogenic $3-loop in any subvariety $3 of the variety of loops and a complete analogue of Theorem 4.1 of [2] is obtained for any one of these subvarieties. Then, using a result similar to those obtained in [5], it is shown that this construction yields uncountably many right neorings with an identity which generates the additive loop of the right neoring. Conversely, every right neoring with an identity which generates its additive loop can be obtained from a free monogenic $3-loop by the above construction. Each of these right neorings has some properties resembling those of the ring of integers. One possible answer is given to the question raised by Bruck concerning the existence of universal right neorings with free additive loop of arbitrary rank. A brief proof is given, using the results of [4; 5], of the cancellation properties of the monogenic universal right neoring. Finally, we discuss briefly the relationship between right neorings and the logarithmetics of Etherington.

It is shown that the rings under the title admit an order-theoretical characterization as in the commutative case studied by Abian. Introduction. Let R be an associative ring equipped with the binary relation (^) defined by xay if and only if xy = x2 in R. In this paper, it is shown that ( ^ ) is an order relation on R if and only if, R has no nilpotent elements i9*0). Conditions on the binary relation (g) in order that R split into a direct product of division rings are then studied in the light of Abian's result (l, Theorem l). Using similar argumentation and using certain subdirect representation of rings with no nilpotent elements, one obtains a complete similarity with the commutative case (yet, no extra complication in the computa- tions). Conventions. R is an associative ring which is, unless otherwise stated, with no nilpotent elements (other than 0). As a result of (2), R can be embedded into a direct product of skewdomains, R—* YLiei £i (that is to say, rings R, having no one-sided divisors of zero). The former embedding is fixed throughout the paper. It is therefore legiti- mate to identify any element x in R with the family consisting of all its projections (xj.e/. Finally, all definitions in (l) are extended (verbatim) to the present case (of a noncommutative ring R) and are freely used throughout.

Recently Cameron and Storvick introduced and studied an operator valued function space integral related to the Feynman integral. The main theorems of their study establish the existence of the function space integral as a weak operator limit of operators defined at the first stage by finite-dimensional integrals. This paper provides a substantial strengthening of their existence theorem giving the function space integrals as strong operator limits rather than as weak operator limits.

1. In a recent paper [2] Reich has made the observation that an example of Koebe given in 1918 [1] does not fulfill its asserted purpose. This example was to show that vanishing measure of the complement did not assure that a slit domain was minimal. Reich proceeded to fill the gap by carrying out the following somewhat more general construction. Let A be a compact perfect nowhere dense set on the x-axis in the z-plane (z=x+iy). Then there exists a compact set S in the z-plane with the properties: (i) A is the projection of S on the x-axis, (ii) S is composed of segments symmetric with respect to the x-axis and points on the x-axis, at least one segment being present, (iii) any point in S, not on the x-axis and not at the end of a segment in S, is the limit, both from the left and right, of points of S. Once this construction is performed the desired examples are easily given [2, ?4]. The object of the present paper is to give an alternative construction which is very explicit and direct.

In [4], Phelps showed that in certain function algebras the unit ball is the closed convex hull of its extreme points. The algebra, C(X), of complex valued continuous functions on a compact Hausdorff space, will always have this property. The class of logmodular algebras which have a Gleason part which is total also was shown to have the property. In this paper we give an elementary proof of the first result (a proof which is, in theory at least, constructive). The simplest nontrivial example of a logmodular algebra with a total part is the disk algebra (i.e. the functions continuous on the closed disk and analytic in the interior). For this algebra we show that the extreme points (in fact the exposed points) of the unit ball form a dense subset of the boundary of the unit ball. Let U be the unit ball of C(X). It is well known that q in U is an extreme point of U if I q(x) I = 1 for all x in X. Now if f in U never vanishes, then f is in the closed convex hull of the extreme points; in factf is between two uniquely determined extreme points. We need only observe that for each x in X,f(x) is halfway between two uniquely determined extreme points of the disk and that these points vary continuously with x. Now it is not necessarily the case that each function in U can be approximated by a nonvanishing function. For a counterexample we need only look at h(z) = z in the algebra of continuous functions on the disk. (If If(z)-zj 1/e. Letfi, i = 1, *, N be the retract maps obtained from these rotations. Then I 1/N Efi(z)-z

where the oi are the usual generators of Bn. (See [1; 4].) A textile manufacturer asked the following question: Can one decide whether or not a braid in An is also in Qn? (It is possible to "weave" a braid in Qn while keeping its ends "tied together.") Shepperd [6] solved the above decision problem using the generators and relations of the braid group and subgroups. In this paper we give another distinct solution working in terms of automorphisms of free groups. In particular, we use formulas developed in [3; 4]. The author wishes to thank his teacher, and friend, Professor Wilhelm Magnus who has contributed more than anyone else towards the author's knowledge of mathematics.

This note deals with the correction of a small flaw in the proof of a theorem (and a lemma) in [2]. Also a simpler proof of the theorem in question is given. In the proof of Theorem 2.3 in [2], the statement "that distinct L's cannot belong to a common pure subgroup HL follows immediately from the fact that no two L's generate with G [p] the same subgroup of G" is false as it stands. One has to exercise a little more care in the selection of HL. For example, since the union [xa, Xa]aEA of all the L's (in Lemma 2.1 in [2]) is still linearly independent, there exists according to [1] a pure subgroup H of G supported by S such that HDL for each L. Hence one could choose HL=H for each L. However, all goes well if HL is chosen such that HLn [Xa, XTg aEA =L, as can easily be done. Similar remarks hold for Lemma 3.8 in [2]. A short alternate proof of Theorem 2.3 in [2] is given. In what follows let c be the cardinality of the continuum and Q be the first ordinal whose cardinality is c. The notation and terminology will be the same as that in [2].

M. Rosenblum in [2] presented a most ingenious proof of the Fuglede and Putnam Theorems by means of entire vector valued functions [1, p. 59]. We will demonstrate that some curious properties of bounded Hilbert space operators can be derived from Rosenblum's argument and similar arguments. Throughout this text we mean by an "operator" a bounded linear transformation of a Hilbert space into itself. Given an operator A we mean by "exp A " the uniform limit of the series I+A +A 2/2 1 +A3/3! +A4/4! + * * * . We let A * denote the adjoint of the operator A, and let z* denote the complex conjugate of the complex number z. A "normal" operator is an operator which commutes with its adjoint. A critical fact in the Rosenblum proof is that given a normal operator A and any complex number z, exp (izA) exp (iz*A *) exp (izA +iz*A *) = exp (iz*A *) exp (izA), and this operator is unitary because i(zA +z*A *) is skew hermitian. Our first result states, among other things, that the converse is true; if the above equations hold for a fixed operator A and all complex numbers z, then A is normal.

The incorrectness of Birkhoff's result, with which the paper [2] is concerned, was noted already by F. R. Gantmacher in 1954 in his book Theory of matrices (Russian); cf. the recent English translation of the second part of this, [l, pp. 175-176]. Example C in [2] is, in fact, the analogue of Gantmacher's counter-example when z= a> is taken instead of 2 = 0 as the singular point. Gantmacher does not, however, discuss the cases in which the result is correct; cf. [2, D]. Unfortunately, the writer became aware of Gantmacher's work only when it was too late to have paper [2] withdrawn or even to have a note added to it.

F(k) ={xkj x F}; F,=F; Fk= (Fk_l,F). Let (u, v, 0)=u, (u, v, 1)=(u, v), (u, v, n)=((u, v, n-1), v). Then Sanov [3] proved that (1. 1) (u, v, apa _1) p-a E F(p#)Fapa+?, /3 a = 1, 2, where p is a prime. In this paper, (1.1) is proved for the cases a= 1, 2; 3 arbitrary. A slight generalization of these results is also proved. Sanov's proof involved an investigation of ideals in a Lie Ring. In this paper, Hall's Collection Process will be used. The method also yields other formulas, e.g. (1.2) (u, v, p2 1)P' (E F2p2_pF(p) (1.3) (u, v, pa+l 1)p~l C F2pa+l paF(pl), a = 1 2, and can be used to produce numerous formulas of a similar nature. The author hopes that some of these formulas and/or the method may be of use in solving other group-theoretic problems. The author was unable to use the method to prove (1.1) for a =3. Note that for a=f = 1, (1. 1) becomes (1.4) (U, v, p 1) E F(p)Fv+1. (1.4) plays an important role in the theory of the Restricted Burnside Problem.

If BP denotes the space of smooth alternating p-forms on the C' manifold M, we are interested in finding the spaces of R-linear maps from iPP to be that commute with the diffeomorphisms of M. For a compact manifold M these spaces were found by R. S. Palais. In this note we find them for noncompact M. Let V, W be the spaces of sections of two tensor bundles over a connected Coo manifold M of dimension n. In a paper [3] appearing in Trans. Amer. Math. Soc. 92 (1959), R. S. Palais studied a number of the spaces 4 (V, W) of the R-linear maps V->W natural with respect to the action on V and W of the group G of diffeomorphisms on M. We sketch the definition of this action: If g-G we define R,:F(TM) -4F(TM) to be the differential of g, and Rg:r(T*M)-->F(T*M) to be the dual of the differential of g-1. For the other spaces of tensor fields Rg is defined as the obvious tensor product of these two R-linear isomorphisms. A map c: V->W is then called natural if cRg =R,c for all g G. Palais was particularly interested in the case when V and W are the space IP of p-forms and the space bq of q-forms respectively, 0 < p, q

There is a mistake in each of [3] and [4]. The purpose of this note is to correct the error in [3] and to salvage what can be saved in [4]. In each case the notation will be that of the paper under discussion. Professor Homer Bechtell has kindly informed me of an error in the proof of the Theorem of [3] which occurs in the case MG1c G. The theorem is true however by altering the proof at this point. Assume that Mc: MG1=M1lc:G. M1 is an invariant subgroup of G, and since G1c: Soc(G), G1 is a direct sum of minimal invariant subgroups of G. By Clifford's theorem (p. 70 of [1]), each minimal invariant subgroup of G is either M1-central or M1-hypereccentric, and hence

The first part of the theorem of the above paper states that if ois a closed convex set in real X1 ... Xn-space which is invariant under permutations of coordinates, and if C(o-) denotes the set of real symmetric nXn matrices whose eigenvalues X1, X. form the coordinates of points in a, then C(o) is convex. I am obliged to Professor R. T. Rockafellar for pointing out that this statement is essentially contained in a theorem of Chandler Davis.2 The statement also follows from an earlier result of V. B. Lidskii,3 which was not known to me at the time of publication.

C. Hobby and C. R. B. Wright [2] have just published the following Theorem A. However, their proof seems to contain an error.2 The notation of [2] is used except that G. is not reserved for the nth term of the lower central series of G: +/(G) denotes the Frattini subgroup of G; (G, H) means the group generated by the commutators g-lh-1gh where gEG, hEH; (A1, A2, * * *, A.+,) is defined inductively as ((A1, A2, . , A.), A.+,); HCG means that H is properly included in G.

PROOF. By a theorem of Rudin a function gEH' in U whose boundary values are real a.e. on I can be analytically continued to D [3, p. 59]. The lemma follows on applying Rudin's result to gi= (1/2) (fl+f2) and g2=(i/2) (fi-f2). PROOF OF THEOREM 1. By a well-known decomposition theorem [2, p. 87], f(z)=B(z)F2(Z), where B(z) is a Blaschke product and F(z) EH1. Since the boundary values of B (z) have absolute value one a.e. on K, we have a.e. on I, f(ei0)= |f(eio)I, or B(ei0)F2(ei0) = F2(ei0) |, and hence

must hold. By an elementary theorem of geometry, C3 /t2s3M3(s) = 1, and a similar argument shows that tc satisfies this same relation in the case where wx/2 1. Because of the restriction on tc only those values of s may be used for which sM(s) oo as s->1, it is not permissible to let s-*1. Added in proof. See review by E. Reich, Math. Rev. vol. 19 (1958) p. 736.

A p-ring (or generalized Boolean ring) P is a ring of fixed prime characteristic p in which ax=a for all a in P. In this paper P is partially ordered by a relation which is a generalization of the usual Boolean order. A subset S of P is then called quasiorthogonal if ab(a-b)=O for all a, b in S. It is shown that P is injective in the category of p-rings if and only if every quasiorthogonal subset has a supremum under this partial order. Sikorski [4] has shown that in the category 9 of Boolean rings the injective objects are the complete Boolean rings. The purpose of this paper is to present a generalization of this result to the category 9 of p-rings, where 9 is understood to be the category with objects rings P of fixed prime characteristic p in which aP=a for all a e P and with morphisms the usual ring homomorphisms. If P is a p-ring, then the set B(P) of idempotents of P is a Boolean ring under the multiplication of P and the new addition defined by aE3b=a+ b-2ab. Batbedat [2] has used B to establish an isomorphism between b? and R. (See also Stringall [6].) Hence, the injective objects in 9 are simply those P for which B(P) is a complete Boolean ring. In this paper the irnjective objects in b? will be characterized by a kind of completeness of a particular partial order that is an extension of the usual partial order on the Boolean ring of idempotents. DEFINITION ]. For a, b E P, a_b if and only if aP-1b=a. It is easily shown that _ is a partial order on P (see, e.g., Abian [1]), but in general (P, _) is not a lattice. It is, however, a lower semilattice with meet defined by aAb=a-a(a-b)P-1. A simple calculation shows that a(bAc)=abAac and aAO=O for a, b, c e P. Foster [3] has shown that every element a of P is uniquely expressible as a sum of elements {ej(a): jE Z.} of B(P). In particular, v-1 (1) a = Zjej(a) i=l Presented to the Society, January 26, 1973 under the title Injectivity in the category of p-rings; received by the editors February 27, 1973. AMS (MOS) subject classifications (1970). Primary 06A70; Secondary 06A40, 06A23.

Results connecting binary quadratic forms and their associated quadratic fields are extended to irreducible decomposable forms and their associated fields. A rational linear substitution that carries such a form into a nonzero rational multiple of itself is shown to correspond with a linear map which admits a unique decomposition as multiplication by a nonzero element of the field followed by an automorphism of the field. This correspondence is one-to-one whenever the form is nondegenerate. 1. Introduction. In recent papers by Butts and Pall (3), Butts and Estes (2), and Taussky (4), the representations of a rational multiple cip of binary quadratic form y by another binary quadratic form are studied. In this paper we generalize these results to rational representations of powers of irreducible decomposable forms. We show (Corollary 3.1) that any irreducible decomposable form which does not properly represent zero (for decomposable forms this is equivalent to being nondegenerate) and whose degree is relatively prime to the number of its variables has only a finite number of rational automorphs. Another interesting result (Corollary 1.3) is that a norm preserving linear endomorphism of an alge- braic number field (considered as a linear space over the rationals) which leaves the identity fixed is an automorphism of the field. This result is well known for quadratic fields and (in modified form) quaternion algebras, and provides one of the cornerstones for the theory of compo- sition of binary and quaternary quadratic forms.

For a sufficiently general immersion of a smooth or polyhedral closed 2-dimensional surface into Euclidean 3-space, the number of triple points is congruent modulo 2 to the Euler characteristic. The approach of this paper involves elementary notions of modification of surfaces by surgery. Let f: M2 2E3 be an immersion in general position of a closed surface M2 into Euclidean 3-space so that there are finitely many N(f) triple points of /. The purpose of this note is to give a direct and elementary proof of the fact that N(f) x(M) (mod 2). The techniques used in this paper are related to the notion of surgery for surfaces, by which a pair of discs is replaced by a cylinder having the same boundary. The approach used in this paper was reported on at the American Mathematical Society Annual Meeting in Las Vegas, January 1972. With the added assumption of differentiability, it is possible to give a new proof of this same result using normal characteristic classes and singularities of projections, as in [l]. For an immersion /: M2 -. E3, set Gr(f) = lx in E3 f| -1(x) consists of precisely r points of M2}. The condition that f is in general position implies that if x e Gr(f) and f1(x) = tp1, P2,..., P,i then there are disjoint disc neighborhoods D(p1), D(p2),.9., D(pr) of these points in M and a homeomorphism w: B D3 of a ball neighborhood B of x to the unit ball in E3 so that (w of) (D(pi)) is the intersection of D3 with the plane orthogonal to the ith coordinate vector. If the immersion f is from a particular category, for example, differentiable or piecewise-linear, then we may assume that the homeomorphism w is in the same category, and in fact all the constructions which will be described can be altered in fairly standard ways so Presented to the Society January 18, 1972; received by the editors September 10, 1973. AMS(MOS) subject classfications(1970). Primary 57C35, 57D40, 57D65; Secondary 55A20.

In this paper we show that a ring R is von Neumann regular (or a V-ring) if and only if every central localization of R at a maximal ideal of its center is von Neumann regular (or a V-ring). Strongly regular rings are characterized by the property that all central localizations at maximal ideals of the center are division rings. Also we consider whether regular PI-rings can be characterized by the property that all central localizations at maximal ideals of the center are simple. Commutative von Neumann regular rings have been characterized in various ways. However, very few of these characterizations extend to noncommutative rings. The results in this paper arose from attempting to extend to noncommutative rings a well-known theorem of Kaplansky [11, Theorem 6] which states that commutative regular rings are characterized by the property that all localizations at maximal ideals are fields. It turns out that the obvious extension of this theorem to the noncommutative case is valid. That being, a ring is regular if and only. if all central localizations at maximal ideals of the center are regular. An analogous theorem is obtained for Vrings. Also we show that strongly regular rings are characterized by the property that all central localizations at maximal ideals of the center are division rings. With an eye to the commutative theory, we consider whether regular PIrings can be characterized by the property that all central localizations at maximal ideals of the center are simple. We provide an example to show that this is not the case. However, it is true if and only if contraction provides a 1:1 correspondence between maximal ideals of the ring and maximal ideals

In this paper a relationship is established between the concepts of E-continuity and shape for compact metric spaces, thus answering a question raised by Victor Klee. Victor Klee [4] has asked whether there is a relationship between E-continuity and shape. Here we show that, in the category of compact metric spaces, the shape morphisms can be identified in a one-to-one mariner with the limits of certain equivalence classes of c-continuous functions. Throughout this paper all spaces are assumed to be compact metric spaces. If f is a continuous function (map) from one space X to another space Y, [/] will denote the homotopy equivalence class of f. By Cov(Y) we will denote the set of all finite open covers of Y directed by refinement. If A and B are collections of subsets of Y, A 0, if and only if it is a-continuous for some a E Cov(Y) consisting of open balls of radius E). If g is another function from X to Y, f and g are a-homotopic if there is an a-continuous function H from X x I to Y such that, for each x E X, H(x, 0) = f(x) and H(x, 1) = g(x). This defines an equivalence relation on the set of a-continuous functions from X to Y. We say that f and g are a-close if /f(x), g(x) }: x E X I < a. For a E Cov(Y), [X, Y]a denotes the set of a-homotopy classes of a-continuous functions f from X to Y. The equivalence class of such an f is denoted f] a. By lim [X, Y] a we denote the usual representation, in the category of sets and functions, of the limit of the inverse system if X, Y]a P aal a < a E Cov(Y)} where Paa'([f]a') = [f]a for a < a E Cov(Y). Received by the editors June 29, 1973 and, in revised form, November 11, 1973. AMS (MOS) subject classifications (1970). Primary 55D99.

for all x, yER. In this paper we study Lie isomorphisms of a primitive ring R onto a primitive ring R', where we assume that the characteristic of R is different from 2 and 3 and that R contains three nonzero orthogonal idempotents whose sum is the identity. Such isomorphisms will be shown to be of the form o+r, where ois either an isomorphism or the negative of an anti-isomorphism of R into a primitive ring L' containing R' and r is an additive mapping of R into the center of L' which maps commutators into zero. This generalizes a theorem of Hua [1], who obtained the above result in the case that R (= '= L') was the ring of all n X n matrices over a division ring, n> 2. On the other hand, due to our requirement concerning idempotents, we fall far short of providing a general solution to Herstein's conjecture [2] that the result holds for arbitrary simple rings. An important part of our proof consists of a repetition of arguments involving matrix units used by Hua [1], and for the sake of completeness (and also because of the relative inaccessability of Hua's paper) we shall reproduce his proofs in some detail when the occasion demands. The author is especially grateful to Professor Nathan Jacobson for several valuable comments towards improving an earlier version of this paper. In the case of simple rings he indicated how all calculations involving matrix units can be eliminated, and he suggested extending our original result to the more general case of primitive rings.

In this paper a study is made of continuous collections of decomposable continua on a spherical surface. Properties of the decomposition spaces of such collections filling up continua are established, and a characterization of the decomposition spaces of such collections filling up a spherical surface is obtained. The results of the present paper are related to certain results obtained by R. D. Anderson. He has shown [2 ] that there is a continuous collection of nondegenerate continua filling up the plane which is with respect to its elements homeomorphic to the plane, and that there is a continuous collection of continua filling up a plane onedimensional continuous curve which is with respect to its elements homeomorphic to the plane. On the other hand, he has obtained [1 I a characterization of the plane decomposition spaces of continuous collections of nondegenerate continuous curves filling up a plane continuum. Such decomposition spaces are special types of hereditary continuous curves. The results of the present paper show that the decomposition spaces of continuous collections of decomposable continua filling up compact plane continua are special types of hereditary continuous curves, and consequently that even under this weaker hypothesis such collections are dimension reducing if they fill up twodimensional continua and are dimension preserving if they fill up one-dimensional continua. It might be noted that there do exist such collections filling up one-dimensional continua, but that Theorem 2 of this paper gives a strong restriction about the nature of such collections. Theorems about continuous collections of continua in a compact metric space can be stated equivalently in terms of monotone interior transformations (p. 130 of reference [10]). It is understood throughout this paper that space is compact and metric. DEFINITIONS. If H is a point set and e is a positive number, V(e, H) denotes the set of all points P such that some point of H is at a distance from P of less than e. If H and K are point sets, d(H, K) will be used to denote the greatest lower bound of the set of all positive numbers e such that K intersects V(e, H). For two

In my dissertation' I defined a W. set as follows: If n > 1, a W. set is a compact continuum M for which there exists a family F of n elements such that (1) each element of F is an upper semicontinuous collection of mutually exclusive continua which fills up M and is an arc with respect to its elements, and (2) if G is a collection of continua each belonging to some, but no two to the same, collection of the family F, then the continua of the collection G have a point in common and their common part is totally disconnected. In that paper it was stated2 without proof that in the plane there exists a W3 set M whose boundary is the point set B(M) and which has a complementary domain whose boundary, J, contains six limit points of B(M) J but that no W3 set has a complementary domain whose boundary contains seven such points. It is the purpose of this paper to prove this statement. In what follows the space considered will be the plane and if M is a point set the notation B(M) will be used to denote its boundary.

This paper constructs a basis for C([O, 1]) which converges weakly to zero whose elements are nevertheless norm bounded away from zero. Introduction. A basis for a Banach space X is a sequence {x, } CX such that to each yEzX there exists a unique sequence of scalars cn for which the partial sums of E CnXn converge to y in the norm of X. Bases can be distinguished by combinatorial and topological criteria (cf. [4]). Foiaa and Singer asked in [1] if C([0, 1]) possesses a basis of type wco, that is, a basis that converges weakly to zero whose elements have norms bounded away from zero. This paper constructs such a basis using a scheme that may be useful in looking for other peculiar bases in C( [0, 1]). J. R. Holub has recently [2] constructed a basis in co that is weakly closed yet intersects every weak neighborhood of zero defined by a single linear functional. The referee has pointed out that Holub's example and the construction below easily imply the existence of such a basis in C( [0, 1]). A method of construction. A sequence of piecewise linear functions which form a basis in C([0, 1]) can be constructed in the following way. The sequence of functions will be accompanied by a parallel sequence of points in [0, 1]. These points will be called joints. The joints are to mark the intervals on which the functions are actually linear. The functions and joints will be grouped into successive stages, each stage having some finite number of functions and an equal number of joints. The kth function (joint) of the nth stage will be denoted fn,k (an,k). To simplify notation an,k will sometimes be written a(n, k). To form a basis thef.,k must be linearly ordered. Do this by sayingfi,j precedesfn,k if i

In recent years several papers [1; 3; 4; 5; 6; 7; 9; 10; 11] have appeared dealing with the problem of "Factorisatio numerorum," the number f(n) of representations of an integer n as an ordered product of factors greater than 1. As a result, the basic combinatorial properties of f(n) and the asymptotic behavior of its summatory function are well known. In this paper, I determine the asymptotic behavior of f(n) itself for squarefree n and use the result to determine a "normal" order of f(n) for all n. If n is written as a product of powers of distinct primes pi, then f(n) is evidently a symmetric function of the exponents of the pi's. If n is squarefree, all the exponents are 1 and f(n) can be considered a function of the single variable r, the number of distinct prime factors of n. It is therefore convenient to define, for positive integral r

In the author's paper [1 ] it is shown that if a sequence of orientable n-dimensional generalized closed manifolds (abbreviated n-gcm) converge (n-1)-regularly to an n-dimensional set, then the limit set is also an orientable n-gcm (see Definitions 1 and 2). In this paper a similar result is obtained for manifolds with boundary. Throughout the paper we assume that our sets are imbedded in a compact Hausdorff space S and that the cycles are Cech cycles with coefficients in an arbitrary field which we will omit from our notation for a cycle. All of the basic homology theory needed is in [7] and a knowledge of it will be assumed. We shall use the notation { A } --+A to mean that the sequence of sets A iCS converges to A CS as a limit (see p. 10 of [5]). We shall consider convergence only when all of the sets Ai are closed, and shall not explicitly state this henceforth; the limit set, as is well known, will always be closed. We shall use small latin letters for points, large ones for sets, and script letters for finite collections of open sets which are coverings of S. We shall use "U" for point set union or sum, "n" for intersection, reserving + and - for the group operations. By VAA we shall mean the subcomplex of the nerve of V whose vertices are elements of the covering V that have a nonvacuous intersection with A. If every element of V is contained in some element of V, we shall say that V is a refinement of V, and shall write V >V, or V = V(V); we shall use the notation' IIH. to denote a simplicial projection from the nerve of V into the nerve of V. By U C C V we mean that UC V, and by V>? we mean that the closure of the elements of V form a refinement of V, and by V> *V, we mean that the stars of the elements of V form a refinement ofV. If a cell a of a covering V has a nucleus (the intersection of all the open sets representing the vertices of a) that meets a set B, we say that af is on B; if all the cells with nonzero coefficients in a chain Cr ofZ V are on B, we say that Cr is on B. If the nucleus of a is in the open set Q, we say a is in Q. We shall use

Introduction. Let L'(G) be the group algebra of an arbitrary locally compact abelian group G. It is a regular semisimple self-adjoint commutative Banach algebra with the LI-norm. The space of regular maximal ideals of L'(G) can be identified with the dual group (character group)P=6 of G. The Gelfand transform of a function f in L'(G) coincides then with the Fourier transform] Off. A subalgebra A of L'(G) is called nondetermining if A is not uniformly dense in Co(r) (cf. [2]). It is called irreducible if it does not contain any closed ideal of L'(G) with hull contained in proper closed subgroup of r. In case A does not contain any ideal of L'(G) at all, it is called completely irreducible. In this paper, we will first show that there is a reduction of maximal nondetermining subalgebras to irreducible ones. The reduced algebra will be a closed subalgebra of L'(G') where G' is a quotient group of G. Next we proved that for certain maximal nondetermining subalgebra A, we have S(A) =) c M(A). Finally, we exhibit a maximal subalgebra of L'(Z??) which is nondetermining and which does not belong to the two categories of maximal subalgebras previously known for discrete locally compact abelian groups, namely, a maximal subalgebra of L'(Z*) which is neither associated with an order of G nor gotten from an essential maximal subalgebra of C(E) for a suitable Cantor set E in r. Throughout this paper, we adopt the terminology and notations of Rudin [4] and Hoffman & Singer [3].

Let d be a divisor of a positive integer n. Then d is a unitary divisor if d and nld are relatively prime, and d is a bi-unitary divisor if the greatest common unitary divisor of d and nld is 1. An integer is bi-unitaty perfect if it equals the sum of its proper biunitary divisors. The purpose of this paper is to show that there are only three bi-unitary perfect numbers, namely 6, 60 and 90. A divisor d of an integer n is a unitary divisor if d and nld are relatively prime. A divisor d of an integer n is a bi-unitary divisor if the greatest common unitary divisor of d and nld is 1. Let a(n) be the sum of the divisors of n, let ?*(n) be the sum of the unitary divisors of n, and let ?**(n) be the sum of the bi-unitary divisors of n. We say that N is unitary perfect if a * (N) = 2N. Subbarao and Warren [2] showed that 6, 60, 90 and 87360 are the first four unitary perfect numbers; Wall reported [3] that 146,361,946,186,458,562,560,000 = 2183.547.11.13.19.37.79.109.157.313 is also unitary perfect and later showed [4] it to be the next such number after 87360. Subbarao [1] has conjectured that there are only finitely many unitary perfect numbers. We say that N is bi-unitary perfect if a* * (N) = 2N. The purpose of this paper is to show that the first three unitary perfect numbers, i.e., 6, 60 and 90, are the only bi-unitary perfect numbers. One easily verifies that ov* is multiplicative and that if p is prime and e> 1, then rY**(pe) = ry(pe) = (pe+l l)/(p 1) if e is odd, and or**(pe) = (pe+l l)/(p 1) -pe/2 if e is even. Hence a**(n)?a(n) with equality if and only if every prime which divides n does so an odd number of times. It also follows immediately that a* * (n) is odd if and only if n is 1 or a power of 2; consequently, each odd prime power unitary divisor of n contributes at least one factor 2 to a**(n). Received by the editors June 10, 1971. AMS 1970 subject classifications. Primary lOA20; Secondary lOA99.

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.

In this paper, a theorem on the centralizer of a closed subgroup H of a Lie group G such that G/H admits a finite invariant measure is proved. Let G be a connected Lie group and H a closed subgroup of G such that the homogeneous space G/H of left cosets admits a positive invariant measure M. The main purpose of this paper is to prove the following theorem. THEOREM. If ,(G/H) is finite, then the closure of the commutator subgroup of the centralizer of H in G is compact. Motivated by the density theorem of Borel [1], Wang [6] has obtained the above-mentioned result, when H is discrete. In ?1, we collect some of the basic facts on measure theory. ?2 carries the complete proof of the announced theorem. Finally, I would like to express my gratitude to Professor T. S. Wu for a useful conversation during the preparation of this paper. 1. Invariant measures on homogeneous spaces. In this section, we summarize some measure theoretic results which will be needed later. For more details, see [4] and [7]. For a locally compact space X, let Cc(X) denote the space of all real valued continuous functions on X with compact support. Then a positive measure on X is a real valued linear functional ,u: CC(X) -*R, such that f Cc(X) is nonnegative; then ,u (f) _ O. Suppose now that G is a locally compact topological group and H is a closed subgroup of G. Given any jp CC(G/H) and aE G, let a(p be defined by a.p(xH) =(p(axH), xHEG/H. Then a positive measure , on G/H is called relatively invariant (relative to the multiplier X) if (1) u,(p) =X(a)Mu((p) for all a G and (p CC(G/H), and (2) X is a homomorphism of G into the multiplicative group R+ of positive real numbers. Received by the editors February 9, 1970. AMS 1969 subject classifications. Primary 2210, 2250; Secondary 2290.

In a recent paper [4]2 it is shown that lattices and trees have a common generalization. These systems will be called median semilattices. These semilattices are characterized below by means of segments (??2 and 4) and by means of betweenness (??3 and 4). In ?5 distributive lattices are characterized by means of medians with no assumption made as to the existence of universal bounds (compare Problem 66 of [1]). In another paper [5], median semilattices are characterized in terms of a binary operation and it is shown that these systems can be imbedded in distributive lattices.

In a previous paper [7], we studied the structure of HID spaces. In this paper, we consider the behavior of HID spaces under monotone mappings. The principal result of this paper is that, given an arbitrary compact metric space Y, there is an HID space X and a monotone map f: X-) Y. We also show that an arbitrary HID space can be mapped monotonically onto a space of any preassigned dimension, and that, given an HID space X and a positive integer n, there is an n-dimensional space Y and a monotone map f: Y->X. R. H. Bing showed in [2 ] that each nondegenerate monotone image of a pseudo-arc is a pseudo-arc. The results of this paper show that no similar monotone invariance property holds for spaces of dimension greater than 1. In this paper, all spaces will be compact metric spaces (compacta). We will be dealing with the Hilbert cube, which we regard as being the product of a countably infinite collection of straight line intervals IX = 11 X 12 X 13 X ... , where Ij = [-1/2i, 1/2'].

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.

Let P ( H ) P(H) denote the projection lattice of a separable Hilbert space H. For each x ∈ H {\text {x}} \in H , let P x {P_{\text {x}}} denote the projection onto the one dimensional subspace generated by x. If B is a Boolean sublattice of P ( H ) P(H) , then it is a theorem that whenever B is maximal in P ( H ) P(H) there exists a vector x 0 ∈ H {{\text {x}}_0} \in H , called a cyclic vector for B, such that the join in P ( H ) P(H) of all the P Q ( x 0 ) {P_{Q({{\text {x}}_0})}} as Q ranges through B is the identity operator I. In this paper we show that this theorem is an immediate corollary of a more general theorem in orthomodular lattice theory. In addition, a final theorem in the paper makes clear the necessity for the separability assumption on H.

Introduction. In a recent paper, [l], A. Beurling has shown that the positive translates of an integrable function defined on [0, m0) generate, in a certain sense, at least one exponential of the form e-iz, x > O, Ia < 0, provided that the function does not vanish outside a finite interval. It is the converse problem with which we shall be concerned here; namely, to what extent can the exponentials so generated be used to approximate the given function. We are able to give what amounts to a complete solution. The situation resembles strongly that of Schwartz' theory of mean periodic functions [2]. M. Kahane has shown in [3] (see also [41) how this theory can be presented very simply using the notion of Fourier transform of a mean periodic function. Beurling also made use of this method in the present case; however, we shall find it convenient to exploit this tool more systematically, in closer analogy with Kahane's work. We shall also study our approximations in a topology (the same as the one used in the theory of mean periodic functions) which is simpler than that of Beurling. Beurling based his work mainly on a certain division theorem which states roughly that an entire function is of finite exponential type if it is bounded on a half plane and equal to the ratio of two bounded analytic functions on the complementary half plane. The conclusions we make here follow from a refinement of this given in ?3 which yields an upper bound for the type of such an entire function. It should be remarked that B. Nyman ([7, pp. 28-29]) has established a result similar to the one given here, using, however, a quite different topology. (The referee calls attention to this in his report; although I have since had the opportunity to consult Nyman's work, it was not accessible to me in New York at the first writing of this paper.)

Three results are proved concerning the Hausdorff fractional dimension of sets of continued fractions whose partial quotients belong to given sequences. Introduction. In [1], I. J. Good investigated the fractional dimension of sets of continued fractions whose partial quotients an obey various conditions. Included amongst these results are theorems discussing cases where an becomes large, and in [2] these results were extended to cover some cases where an tends to infinity rapidly. In all these results the only restrictions on an are of the type an_f(n) and an--oo. In this paper I shall prove analogous results concerning the cases where an is further restricted to belong to some sequence of natural numbers. The notation to be used and the relevant parts of the theories of continued fractions and Hausdorff measures are given in [2] and the reader is referred to that paper for these details. In addition throughout the paper we shall use (bn) to denote a strictly increasing sequence of natural numbers (rather than (+(n))-for ease of printing). The following theorems will be proved. THEOREM 1. Suppose the series E (on)-' converges. Then provided A has the property that >,(on)" -A the set E={jai.>A and ai E (bn)} has fractional dimension _ ?cx. THEOREM 2. Suppose the series E (bn)-' diverges. Then for any A, the set E in Thleorem 1 has dimension 1ac.

In [2] Albert proved that a finite-dimensional absolute-valued algebra over the reals is necessarily aiternative (and hence the reals, complexes, quaternions, or Cayley numbers). In [3 ] he extended this from finite-dimensional to algebraic algebras. Recently, Wright [9] succeeded in removing the assumption that the algebra is algebraic. Wright proceeds by proving that the norm springs from an inner product, and then that the algebra is algebraic. Now if the norm I xI comes from an inner product, then xI 2 is a quadratic form in x, and moreover the assumption xy = x I y| means that it is a quadratic form admitting composition. Thus Albert's finite-dimensional theorem can be proved by combining Wright's result with Hurwitz's classical theorem on quadratic forms admitting composition. The main purpose of this paper is to make a similar method possible in the infinite-dimensional case by providing a suitable generalization of Hurwitz's theorem. The result is essentially that infinite-dimensional quadratic forms cannot admit composition, except in the rather trivial case of purely inseparable fields of characteristic two. In the concluding moments of the proof we rely heavily on the recent developments in the theory of alternative rings. Until then the paper is quite self-contained and elementary, and the style of the argument is very close to that of [1], [5], and [8]. Let A be a vector space over a field F. The function g(x) from A to F is a quadratic form if

The purpose of this paper is to give a necessary and sufficient condition in order that the S limit set of a point consists of exactly one periodic orbit which is both S-Lyapunov stable and S-asymptotically stable. The general reference for most terms used in this paper is [2]. In this paper we assume (X, T, r) is a transformation group. X will always be assumed to be a uniform space where the Hausdorff topology in X is induced by the uniformity. T will be assumed to be generative, that is, T is isomorphic to Rm X I X C where R is the additive group of real numbers, I is the additive group of integers, m and n are nonnegative integers, and C is a compact abelian group [4]. S will denote a closed replete semigroup in T. The S-limit set of yEX, Sy, is defined bySv = n te,s Cl (ytS) [1 ]. DEFINITION 1. The orbit of a point yEX is said to be uniformly S-Lyapunov stable with respect to BCX provided that for each index a of X there is an index ,3 of X such that if weEBnyt'13 for some t' E T then wt Eyt'ta for all t E S.

This paper constructs a minimal element in the partial order on the set of skew fields generated by a free algebra, and shows that the partial order contains a certain sub partial order. Examples of embedding free algebras in skew fields of heights one and two are also given. Let R be an integral domain (noncommutative) with identity. We will assume that the integral domain R is always embedded in some skew field (this need not be the case in general [2], [3], [6], [7]). There are many skew fields which contain R but in the commutative case there is only one field which is generated by R, namely the field of fractions. In the noncommutative case there may be several distinct skew fields generated by R [4, pp. 277]. For skew fields D1 and D2 generated by R, we say D1 > D2 if there exists a place from D1 to D2 which extends the natural isomorphism between the embeddings of R. This paper shows that this is a partial order on the set (P (we identify isomorphic embeddings) of skew fields generated by R and in the case where R is the free algebra on two generators we show that (P contains the subposet ? where C(x) is the unique maximal element [1, Theorem 27 ] of (P and Ki are distinct elements of (P with K2 minimal in (P. We also examine the height of an integral domain and give examples of embeddings of different heights of a free algebra in two skew fields. I. The partial order and the chain of domains Qi(K, ?(R)). DEFINITION. Let D be a skew field and 0 an isomorphism of R into D. R is fully embedded in D if the smallest sub skew field of D containing + (R) is D itself. We will denote a full embedding of (D, 4 (R)) by (D, q). DEFINITION. If D1 and D2 are two division rings we say 0 is a place from D1 to D2 if 0 is a homomorphism from a local subring S of D, onto D2 [local means the set of nonunits is an ideal ]. DEFINITION. For full embeddings (D, a) and (K, y) we say (D, a) > (K, Py) if there is a place q from D to K such that qV'(K) Doa(R) and f1 a (R) =,y-c'. Received by the editors October 9, 1970. AMS 1969 subject classifications. Primary 1646, 1615.

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