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Erdös [1] has shown that, if the square-free numbers in ascending order be denoted by s1, s2, … , sn, … , then for 0 ≦ γ ≦ 2as x → ∞. In this paper we shall extend this result by proving that the asymptotic formula in fact holds for the wider range 0 ≦ γ ≦ 3.Similar results have been obtained previously by the author in respect of both the sequence of numbers expressible as the sum of two squares and also sequences of numbers relatively prime to given large integers, although the method used here differs from that of the earlier papers [2; 3].

A dissectible polyhedron is a natural extension of a concept whose history dates back to at least 1758 and Euler [7]—the concept of a dissection of a polygon. An interesting historical survey of dissections of a polygon is given by Brown [4]. Some approaches to the classical problem have been given by Moon and Moser [9] and by Guy [8] ; the latter provides an approach which is the basis of the work in this paper. A summary of enumeration results on dissections of polygons and polyhedra by automorphism groups has been given by the authors [2].Recent extensions of the problem have been investigated in a series of papers by Brown and Tutte [3; 5; 14; 15] and b y Takeo [10; 11; 12; 13].

Introduction. Until recently none of the numerous papers on the distribution of quadratic and higher power residues was concerned with questions of the following sort: Let k and m be positive integers. According to a theorem of Brauer (1), for every sufficiently large prime p there exist m consecutive positive integers r, r + l , . . . , r + m — 1, each of which is a &th power residue of p. Let r(k, m, p) denote the least such r. What can be said about the behaviour of r(k, m, p) as p varies? In particular, for what values of k and m is r(k, m, p) bounded, and for these values what is its maximum? For a fixed k and m we call a prime exceptional and denote it by p* if there do not exist m consecutive integers each of which is a &th power residue of p*. Set A(k,m) = Max r (k, m, p), where the maximum is taken over all nonexceptional primes p. In a previous paper (4) Lehmer and Lehmer showed that

By an Euler polyhedron of valence three or a trivalent convex polyhedron in Euclidean 3-space (4) we mean in the present paper an Euler polyhedron in the sense of Steinitz (8, p. 113), such that each vertex is incident with exactly three edges.In the present paper we establish a theorem concerning the colouring of trivalent polyhedra. A specialization of this theorem solves the following problem implicit in Eberhard (1, p. 84): Does there exist a trivalent Euler polyhedron with an odd number of faces such that the number of edges incident with any face is divisible by three?

If G is a finite group which has a faithful complex representation of degree nit is said to be a linear group of degree n. It is convenient to consider only unimodular irreducible representations. For n ≦ 4 these groups have been known for a long time. An account may be found in Blichfeldt's book (1). For n= 5 they were determined by Brauer in (4). In (4), many properties of linear groups of prime degree pwere determined for pa prime greater than or equal to 5.In a forthcoming series of papers these results will be extended and the linear groups of degree 7 determined. In the first paper, some general results on linear groups of degree p, p≧ 7, will be given. These results will later be applied to the prime p = 7.

We say a (plane) set A contains all sets of some type if, for each B of type , there is a subset of A that is congruent to B. Recently, Besicovitch and Rado [3] and independently, Kinney [5] have constructed sets of plane measure zero containing all circles. In these papers it is pointed out that the set of all similar rectangles, some sets of confocal conies and other such classes of sets can be contained in sets of plane measure zero, but all these generalizations rely in some way on the symmetry, or similarity of the sets within the given type.In this paper we construct a set of plane measure zero containing all finite polygonal arcs (i.e., the one-dimensional boundaries of all polygons with a finite number of sides) with slightly stronger results if we restrict our attention to k-gons for some fixed k.

In a recent paper by Douglas Underwood [8] several definitions of “associated prime divisors” were discussed and shown to be unique. In this note we produce a fifth type, which is due to W. Krull, and is found in his classical paper [2] and further discussed by B. Banaschewski in[1].Historically this characterization considerably predates the other four definitions.Throughout this note,Rdenotes a commutative ring with unity, and all ideals and elements are assumed to be in such a ring. We shall let upper case letters, most frequently the beginning of the alphabet, denote ideals and lower case letters, elements ofR.On the whole, our terminology will be that of [9]. We do, however, take exception with [9] in two instances, viz.

The existence of certain formulae analogous to Poisson's summation formula (9, pp. 60-64),where αβ = 2π, α > 0, and Fc(x) is the Fourier cosine transform of f(x), but involving number-theoretic functions as coefficients, was first demonstrated by Voronoï (10) in 1904. He proved thatwhere r(n) is an arithmetic function,/(x) is continuous in (a, b) and a(x) and i?(x) are analytic functions dependent on τ(n). Later, numerous papers were published by various authors giving formulae of this type involving d(n), the number of divisors of n (3), and rp(n), the number of ways of expressing n as the sum of p squares of integers (8).In 1937, Ferrar (4) developed a general theory of summation formulae, using complex analysis. Around that time, Guinand (5) also published papers where he developed the general theory from a different point of view. He applied the theory of mean convergence for the transforms of class L2(0, ∞ ). Later in 1950, Bochner (1) gave a general summation formula.

In two previous papers (see4;5) O. T. O'Meara and I investigated the problem of generating the integral orthogonal group of a quadratic form by symmetries in the case where the underlying ring of integers was the integers of a dyadic local field of characteristic not 2. In this paper, the investigation is concerned with a local field of characteristic 2. As in (5), only the unimodular case is treated. As in (4) and (5), groupsS(L), Xh(L), andO(L) are introduced for a unimodular latticeLand the relationship betweenS(L) andO(L) studied. As in the previously cited papers, generation by symmetries means thatS(L) =O(L). The following result is obtained.

Error-correcting codes are used in several constructions for packings of equal spheres in ^-dimensional Euclidean spaces En. These include a systematic derivation of many of the best sphere packings known, and construction of new packings in dimensions 9-15, 36, 40, 48, 60, and 2m for m g 6. Most of the new packings are nonlattice packings. These new packings increase the previously greatest known numbers of spheres which one sphere may touch, and, except in dimensions 9, 12, 14, 15, they include denser packings than any previously known. The density A of the packings in En for n = 2m satisfies log A ~ — \n log log n as n —* oo. 1.1. Introduction. In this paper we make systematic use of error-correct ing codes to obtain sphere packings in En, including several of the densest packings known and several new packings. By use of cross-section s we then obtain packings in spaces of lower dimension, and by building up packings by layers we obtain packings in spaces of higher dimension. Collectively, these include all of the densest packings known, and further new packings are also con­ structed. Part 1 of the paper is devoted to groundwork for the constructions. § 1.2 introduces sphere packings, and §§ 1.3-1.8 survey the error-correcting code theory used in the later Parts. Part 2 describes and exploits Construction A, which is of main value in up to 15 dimensions. Part 3 describes Construction B% of main value in 16-24 dimensions. Part 4 digresses to deal with packings built up from layers, while Part 5 gives some special constructions for dimen­ sions 36, 40, 48 and 60. Part 6 deals with Construction C, applicable to dimensions n = 2m and giving new denser packings for m ^ 6. We conclude with tables summarizing the results. Table I, for all n S 24, supersedes the tables of [18; 19], and Table II gives results for selected n > 24. The tables may be used as an index giving references to the sections of the paper in which the packings are discussed. Partial summaries of this work have appeared in [22; 23]. General references for sphere packing are [18; 19; 31] and for coding theory [4; 25].

I t is straightforward, but tedious, to write down the integers whose representations in a given base do not have particular digits in certain positions. In the first section of this paper we give a computational scheme that enables us to carry out such operations in a rapid and simple fashion.In the second section of the paper we derive a general identity involving the digits of integers in arbitrary Cantor systems of notation.In the third section we apply this identity and deduce a number of results concerned with the splitting of integers into classes with equal power sums. The computational scheme of the first section leads us to an algorithm for the determination of such splittings.

LetRbe the discrete group ring of the groupGover the ringA. In this paper we attempt to find necessary and sufficient conditions onGandAso thatRwill have some standard ring-theoretic property ; among the properties considered are those of being artinian, regular, self-injective, and semi-prime.The contents of this paper form essentially the author's doctoral thesis. The author would like to thank his supervisor Dr. J. Lambek for his generous encouragement and continued interest.

Let Mm be a closed PL manifold of dimension m. Then a concordance between two PL-homeomorphisms h0, h1:M → M is a PL-homeomorphismH: M × I → M × I such that H|M × 0 = h0 and H|M × 1 = h. Concordance is an equivalence relation and in his paper [2], M. Kato classifies PL-homeomorphisms of Sp × Sq up to concordance. To do this he treats first the problem of classifying those homeomorphisms that induce the identity in homology, and then describes the automorphisms of the cohomology ring that can arise from homeomorphisms of Sp × Sq. In this paper we show that for sufficiently connected PL-manifolds that embed in codimension 1, one can extend Kato's classification of the homeomorphisms that induce the identity in homology.

For a formal definition of “map” the reader is referred to (7, §2). The maps in this paper are rooted by specifying an orientation for one of the edges. This also specifies a root vertex, the negative end of the root, and a root face, the face on the left of the root edge. Counting is, as usual, defined on isomorphism classes.Regular maps of even valence have been enumerated in a recent paper by Tutte. In this paper we determine the average number of trees in such maps, and include similar results for regular tri valent maps, that is, maps with three edges incident on every vertex. In the development for the latter, a formula for the number of trivalent maps with 2t vertices is produced.

HNN groups have appeared in several papers, e.g., [3; 4; 5; 6; 8]. In this paper we use the results in [6] to obtain a structure theorem for the subgroups of an HNN group and give several applications.We shall use the terminology and notation of [6]. In particular, if K is a group and {φi} is a collection of isomorphisms of subgroups {Li} into K, then we call the group1the HNN group with base K, associated subgroups { Li,φi(Li)} and free part the group generated by t1, t2, …. (We usually denote φi(Li) by Mi or L–i.) The notion of a tree product as defined in [6] will also be needed.

Throughout this paper, Z is the ring of integers, ƒ*(t) (ƒ(t)) is an integer monic (co-monic) polynomial in the indeterminate t (i.e., each coefficient of ƒ* (ƒ) is in Z and its highest (lowest) coefficient is 1 (5, p. 121, Definition) and M* (M) is the multiplicative semigroup of all integer monic (co-monic) polynomials ƒ* (ƒ) having no constant term. In (3, Theorem 2), Herstein proved that if R is a division ring with centre C such that1then R = C. In this paper we seek a generalization of Herstein's result to semi-simple rings. We also study the following condition:(1)*Our results are quite complete for a semi-simple ring R in which there exists a bound for the codegree ofƒ (ƒ*) (i.e., the degree of the lowest monomial of ƒ(ƒ*)) appearing in the left-hand side of (1) ((1)*).

Soon after the publication of Tutte's paper [5] on m-cages, H. S. M. Coxeter asked in a letter to the author whether one-regular graphs of degree 3 exist. The purpose of the following paper is to show by an example that the answer is in the affirmative.

Throughout this paper, the symbol P = [Pij] will represent the transition probability matrix of an irreducible, null-recurrent Markov process in discrete time. Explanation of this terminology and basic facts about such chains may be found in (6, ch. 15). It is known (3) that for each such matrix P there is a unique (except for a positive scalar multiple) positive vector Q = {qi} such that QP = Q, or1this vector is often called the "invariant measure" of the Markov chain.The first problem to be considered in this paper is that of determining for which vectors U(0) = {μi(0)} the vectors U(n) converge, or are summable, to the invariant measure Q, where U(n) = U(0)Pn has components2In § 2, this problem is attacked for general P. The main result is a negative one, and shows how to form U(0) for which U(n) will not be (termwise) Abel summable.

For various classesQof metric spaces, there are several well-known results characterizing the localn-connectivity of a metric space in terms ofn-ANR(Q)'s. Specifically, we have in mind the results of Kuratowski (13, p. 265) and Kodama (10, p. 79). The main purpose of this paper will be to obtain similar results along these lines for non-metric classesQ.In the last part of the paper we specifyQto be the class of totally normal spaces and characterize the localn-connectivity of ann-dimensional separable metric space in terms of ANR(Q)'s.

1. 1. This paper is the second in a series of papers discussing linear groups of prime degree, the first being (8). In this paper we discuss only linear groups of degree 7. Thus, G is a finite group with a faithful irreducible complex representation Xof degree 7 which is unimodular and primitive. The character of Xis x- The notation of (8) is used except here p= 7. Thus Pis a 7-Sylow group of G.In §§ 2 and 3 some general theorems about the 3-Sylow group and 5-Sylow group are given. In § 4 the statement of the results when Ghas a non-abelian 7-Sylow group is given. This corresponds to the case |P| =73 or |P|= 74. The proof is given in §§ 5 and 6. In a subsequent paper the results when Pis abelian will be given.

These notes are to supplement my paper (4), and should be read in conjunction with it. Both are divided into three parts, and in these notes the section numbers have a further digit added; thus §1.41 here supplements §1.4 of (4). References by section numbers are always to (4) or to the present notes, but references to other papers are numbered independently.The principal results of these notes are the following. New sphere packings are given in [2m], m ⩾ 6, and in [24], which are twice as dense as those of §§1.6, 2.3. Others are given in [2m], m ⩾ 5, with the same density as those of §1.6, but in which each sphere touches fewer other spheres than in the earlier packings.

In a previous paper (2), one of us has derived a formula for the counting series for bicoloured graphs.2 These are graphs each of whose points has been coloured with exactly one of two colours in such a way that every two adjacent points have different colours.In this paper we first enumerate bicoloured graphs without isolated points and connected bicoloured graphs. This leads us to corresponding problems for bicolourable graphs. Such a graph has the property that its points can be coloured with two colours so as to obtain a bicoloured graph.

Our aim is to present some global results about the set of maximal abelian subgroups of the symmetric group Sn. We shall show that certain properties are true for “almost all” subgroups of this set in the sense that the proportion of subgroups which have these properties tends to 1 as n → ∞. In this context we consider the order and the number of orbits of a maximal abelian subgroup and the number of generators which the group requires.Earlier results of this kind may be found in the papers [1; 2; 3; 4; 5]; the papers of Erdös and Turán deal with properties of the set of elements of Sn. The present work arose out of a conversation with Professor Turán who posed the general problem: given a specific class of subgroups (e.g., the abelian subgroups or the solvable subgroups) of Sn, what kind of properties hold for almost all subgroups of the class?

The main theorem of this paper names seven ring classifications which coincide with the class of rings named in the title. Three of them are (1) rings over which every module is rationally complete, (2) left and right perfect rings over which every module is corationally complete, and (3) right perfect rings over which every right module is rationally complete. Corational completeness (introduced in this paper) and rational completeness are generalizations of projectivity and injectivity, respectively. One recalls that a proper subclass of the rings investigated in this paper, the artinian rings with zero radical, is known to have one-sided characterizations in terms of the injectivity (projectivity) of modules. An example proves, however, that the class of rings over which every right module is rationally complete properly contains the rings of the title.

Let X be an n-square matrix with elements in a field F. The permanent of X is defined by1.1where σ runs over the symmetric group of permutations on 1, 2, … , n. This function makes its appearance in certain combinatorial applications (13), and is involved in a conjecture of van der Waerden (6; 9). Certain formal properties of per (X) are known (1), and an old paper of Pólya (12) shows that for n > 2 one cannot multiply the elements of X by constants in any uniform way so as to convert the permanent into the determinant. In a subsequent paper we intend to investigate this problem for more general operations on X.

In this paper we consider undirected graphs, with no edges joining a vertex to itself, but with possibly several edges joining pairs of vertices. The first part of the paper deals with the question of characterizing those sets of non-negative integers d1d2 . . . , dn and {cij}, 1 ≤ i < j ≤ n, such that there exists a graph G with n vertices whose valences (degrees) are the numbers di, and with the additional property that the number of edges joining i and j is at most cij. This problem has been studied extensively, in the general case (1, 2, 9, 11), in the case where the graph is bipartite (3, 5, 7, 10), and in the case where the Cij are all 1 (6).

The term rank p of a matrix is the order of the largest minor which has a non-zero term in the expansion of its determinant. In a recent paper (1), the authors made the following conjecture. If S is the sum of all the entries in a square matrix of non-negative real numbers and if M is the maximum row or column sum, then the term rank p of the matrix is greater than or equal to the least integer which is greater than or equal to S/M. A generalization of this conjecture is proved in § 2.The term doubly stochastic has been used to describe a matrix of nonnegative entries in which the row and column sums are all equal to one. In this paper, by a doubly stochastic matrix, the, authors mean a matrix of non-negative entries in which the row and column sums are all equal to the same real number T.

In this paper we derive a generation procedure for the simple (3-valent) 3-polytopes with cyclically 5-connected graphs. (A graph is called cyclically n-connected if it cannot be broken into two components, each containing a cycle, by the removal of fewer than n edges.) We define three new types of face splitting and we show, in Theorems 16 and 17, that the simple 3-polytopes with cyclically 5-connected graphs are exactly the polytopes obtained from the dodecahedron by these face splittings.

Our main purpose is to enlarge upon the studies of McAdam [9; 10] on the property of going down (GD) for prime ideals in extensions of (commutative integral) domains. Unlike the investigations of McAdam and the earlier work of Krull [8] and Cohen-Seidenberg [4] on GD and the related property of going up (GU), this paper is not primarily concerned with integral extensions. Consideration of more general extensions of domains A ⊂ B is facilitated by the following basic definitions. A prime ideal P of A is unibranched in B if there exists exactly one prime ideal Q of B satisfying Q ∩ A = P.

There have been several recent papers on the subject of the P-hull and the SP-hull of an l-group (lattice-ordered group). The most natural formulation of the concepts was given by P. Conrad in [6]. T. Speed studied P-groups extensively in [11]; his work was motivated by earlier work by H. Nakano and I. Amemiya in a vector lattice setting. A. Vecksler [12] produced the SP-hull for f-rings. The ortho-completion of S. Bernau [2] is a related concept.

In [4], a subgroup theorem for HNN groups was established. The theorem was proved by embedding the given HNN group in a free product with amalgamated subgroup and then applying the subgroup theorem of [3]. In this paper we obtain a sharper form of the subgroup theorem of [4] by applying the Reidemeister-Schreier method directly, using an appropriate Schreier system of coset representatives. Specifically, we prove (in Theorem 1) that if H is a subgroup of the HNN group1

This paper is concerned with the internal structure of Ext(Q, T) where Q is the group of rationals and T a reduced p-primary group of unbounded order. In [1] Irwin, Khabbaz, and Rayna define the splitting length of an arbitrary abelian group A, written l(A), to be the least positive integer n, otherwise infinity, such that A ⊗ . . . ⊗ A (n factors) splits. The concept of splitting length has been induced on Ext(Q, T), see [2; 5].

A collection of subspaces of a Hilbert space is called a nest if it is totally ordered by inclusion. The set of all bounded linear operators leaving invariant each member of a given nest forms a weakly-closed algebra, called a nest algebra. Nest algebras were introduced by J. R. Ringrose in [9]. The present paper is concerned with generating nest algebras as weakly-closed algebras, and in particular with the following question which was first raised by H. Radjavi and P. Rosenthal in [8], viz: Is every nest algebra on a separable Hilbert space generated, as a weakly-closed algebra, by two operators? That the answer to this question is affirmative is proved by first reducing the problem using the main result of [8] and then by using a characterization of nests due to J. A. Erdos [2].

The work presented in this paper was initially motivated by the following question of A. Wilansky: “Is there a smallest FK-space E in which is bounded?” Here FK-space means a complete linear metric space of real or complex sequences x = (xi) upon which the coordinate functional x → xt are continuous for each i (see [10, p. 202]), and An FK-space need not be locally convex, and therein lies the difficulty of the problem since it is easy to see that l1 is the smallest locally convex FK-space.

An enumeration of known simple groups has been given by Dickson [17]; to this list, he made certain additions in later papers [15], [16]. However, with but five exceptions, all known simple groups fall into infinite families; these five unusual simple groups were discovered by Mathieu [21], [22] and, after occasioning some discussion [20], [23], [27], were relegated to the position, which they still hold, of freakish groups without known relatives. Further interest is attached to these Mathieu groups in virtue of their providing the only known examples (other than the trivial examples of the symmetric and alternating groups) of quadruply and quintuply transitive permutation groups.

In this paper we shall study the relations between the ranks of g-variate, discrete-parameter, weakly stationary stochastic processes x, y, and z satisfying the condition1.1and derive from them a characterization for the Wold decomposition and conditions for the concordance of the Wold and the Lebesgue-Cramér decompositions.

This paper presents two results. They are:Theorem 1. Let G be a doubly transitive permutation group of degree nq + 1 where a is a prime and n < g. If G is neither alternating nor symmetric, then G has Sylow q-subgroup of order only q.Result 2. There is no unsolvable transitive permutation group of degree p = 29, 53, 149, 173, 269, 293, or 317 properly contained in the alternating group of degree p.Result 2 was demonstrated by a computation on the Illiac II computer at the University of Illinois.

A Stone lattice is a distributive, pseudo-complemented lattice L such that a* V a** = 1, for all a in L; or equivalently, a bounded distributive lattice L in which, for all a in L, the annihilator a⊥ = {b ∊ L|a ∧ b = 0} is a principal ideal generated by an element of the centre of L, namely a*.Thus it is natural to define an 𝔪-Stone lattice to be a bounded distributive lattice L in which, for each subset A of cardinality less than or equal to m, the annihilator A⊥ = {b ∊ L|a ∧ b = 0, for all a ∊ A} is a principal ideal generated by an element of the centre of L.In this paper we characterize 𝔪-Stone lattices, and show, by considering the lattice of global sections of an appropriate sheaf, that any bounded distributive lattice can be embedded in an 𝔪-Stone lattice, the embedding being a left adjoint to the forgetful functor.

In his paper (3) on finite H-planes, Kleinfeld has defined invariants s and t for an H-plane π as follows: let P and k be any point and line of π such that P is incident with k, then let s be the number of non-neighbour points to P on k and let t be the number of neighbour points to P on k. He has shown that s and t are independent of the choice of P and k, t divides s, π has s2 + st + t2 points (lines), π* has order s/t, and, if t ≠ 1, s ≤ t2. In the case s = t2, π is called uniform and has the property that each pair of neighbour lines (points) has exactly t points (lines) in common.

The results of the present paper can be interpreted (a) in terms of the theory of the representations of the symmetric group, or (b) in terms of the corresponding theory of the full linear group. In the latter connection they give a solution to the problem of the expression of an invariant matrix of an invariant matrix as a sum of invariant matrices, in the sense of Schur's Dissertation. D. E. Littlewood has pointed out the significance of this problem for invariant theory and has attacked it via Schur functions, i.e. characters of the irreducible representations of the full linear group. We shall confine our attention here to the interpretation (a).

Let D be an integrally closed domain with identity having quotient field L. If {Vα} is the set of valuation overrings of D and if A is an ideal of D, then à = ∪ αAVα is an ideal of D called the completion of A. If X is an indeterminate over D and f ∈ D[X], then we denote by Af the ideal of D generated by the coefficients of f. The Kronecker function ring DK of D is defined by DK = {f/g| f, g ∈ D[X], Ãf ⊆ Ag} (4, p. 558); and the domain D(X) is defined by D(X) = {f/g| f, g ∈ D[X], Ag = D} (5, p. 17). In this paper we wish to relate the ideal theory of D to that of DK and D(X) for the case in which D is a Prüfer domain, a Dedekind domain, or an almost Dedekind domain.

In [1], B. Baumslag defined a subgroup U of a group G to be malnormal in G if gug–1 ∈ U, 1 ≠ u ∈ U, implies that g ∈ U. Baumslag considered the class of amalgamated products (A * B; U) in which U is malnormal in both A and B. These amalgamated products play an important role in the investigations of B. B. Newman [13] of groups with one defining relation having torsion. In this paper, we shall be concerned primarily with a generalization of this class.Let U be a subgroup of a group G and let u ∈ U. Then the extended normalizer EG(u, U) of u relative to U in G is defined byif u ≠ 1, and by EG(u, U) = U if u = 1.

In 1938, Frucht (2) proved that for any given finite group G there exists a finite symmetric graph X such that G(X) is abstractly isomorphic to G. Since G(X) is a permutation group, it is natural to ask the following related question : If P is a given finite permutation group, does there exist a symmetric (and more generally a directed) graph X such that G(X) and P are isomorphic (see Convention below) as permutation groups? The answer for the symmetric case is negative as seen in (3) and more recently in (1). It is the purpose of this paper to deal with this problem further, especially in the directed case. In §3, we supplement Kagno's results (3, pp. 516-520) for symmetric graphs by giving the corresponding results for directed graphs.

The following results (9, Exercise 26, p. 10; 1, Theorem 9.2; 8, Theorem III. 1.11) are known.(A) Let R be a ring with more than one element. Then R is a division ring ifand only if for every a ≠0 in R, there exists a unique b in R such that aba = a.(B) Let R be a near-ring which contains a right identity e ≠ 0. Then R is adivision near-ring if and only if it contains no proper R-subgroups.(C) Let R be a finite near-ring with identity. Then R is a division near-ringif and only if the R-module R+ is simple.In this paper we will show that (A) can be generalized to distributively generated near-rings. We also will extend (B) and (C) to a larger class of near-rings.

Consider the following sieve process. Let A(1) be the sequence of integers greater than 1. Let A(n+1) be obtained from A(n) = {a1(n), a2(n), …} by eliminating one element from each of the intervals Ik(n), whereWe let an = an(n) and A = {an} be the sequence of integers that survive the sieve. M. C. Wunderlich (8) has found a necessary and sufficient condition for an ∼ n log n and, in a more recent paper, M. Wunderlich and W. E. Briggs (9) have studied a subclass of the sequences defined above for which an ∼ n log n.

A well-known theorem asserts that if K is a field, a prime ideal in the polynomial ring S = K[X1, … Xn] and d the transcendence degree of S / over K n = rank + d. In the first half of this paper we extend this result to the case of arbitrary commutative noetherian K, as well as giving a purely homological proof of the classical theorem. In the second half we use our first result to compute the analogue of the dimension of the product and intersection of two affine varieties when K is a Dedekind ring.

The objective of this paper is to give some concrete distinctions between positive linear maps and completely positive linear maps on C*-algebras of operators.Herein, C*-algebras possess an identity and are written in German type . Capital letters A, B, C stand for operators, script letters for vector spaces, small letters x, y, z for vectors. Capital Greek letters Φ, Ψ stand for linear maps on C*-algebras, small Greek letters α, β, γ for complex numbers.We denote by the collection of all n × n complex matrices. () = ⊗ is the C*-algebra of n × n matrices over .

In this paper we prove a sharpening and generalization of the following Theorem of Khintchine (4):Let ψ1(q), …, ψnq) be n non-negative junctions of the positive integer q and assumeis monotonically decreasing. Then the set of inequalities1has an infinity of integer solutions q > 0 and p1, … , pn for almost all or no sets of numbers θ1, … , θ2, according as Σψ(q) diverges or converges.

Let Fn be the collection of n-by-n matrices over a field F. For Y in Fn let ΔY be the mapping on Fn given by XΔY = XY — YX. In this paper we study the followingProposition. Let A and B be in Fn and let m be a positive integer. If BΔxm = 0 whenever XΔAm = 0, then B is a polynomial in A with coefficients in F.

Let R be a commutative ring with unity and let G be a group. The group ring RG is a free R-module having the elements of G as a basis, with multiplication induced byThe first theorem in this paper deals with idempotents in RG and improves a result of Connell. In the second section we consider the Jacobson radical of RG, and we prove a theorem about a class of algebras that includes RG when G is locally finite and R is an algebraically closed field of characteristic zero. The last theorem shows that if R is a field and G is a finite nilpotent group, then RG determines RP for every Sylow subgroup P of G, regardless of the characteristic of R.