Showing papers on "Ring (mathematics) published in 1973"
TL;DR: In this paper, a sheaf representation is obtained for pseudo symmetric rings, similar to the cases of symmetric ring, semiprime ring, and strongly harmonic ring.
Abstract: Almost symmetric rings and pseudo symmetric rings are introduced. The classes of symmetric rings, of almost symmetric rings, and of pseudo symmetric rings are in a strictly increasing order. A sheaf representation is obtained for pseudo symmetric rings, similar to the cases of symmetric rings, semiprime rings, and strongly harmonic rings. Minimal prime ideals of a pseudo symmetric ring have the same characterization, due to J. Kist, as for the commutative case. A characterization is obtained for a pseudo symmetric ring with a certain right quotient ring to have compact minimal prime ideal space, extending a result due to Mewborn. Introduction. Recently Koh [91 has obtained a sheaf representation of a ring without nilpotent elements. While Lambek [121 has unified this and the commutative case by introducing symmetric rings, Hofmann [7, Theorems 1.17 and 1.241 has extended the representation to semiprime rings. Using the maximal modular ideal space, Koh [11] has also obtained the representation for strongly harmonic rings. In this paper, the result of Lambek [121 is extended to a larger class of ringspseudo symmetric rings (Theorem 3.5). Example 5.1(e) is an example of a pseudo symmetric ring whose representation does not fall under any other types mentioned above. (See [7, p. 3111.) Almost symmetric rings are also introduced. A symmetric ring is almost symmetric and an almost symmetric ring is pseudo symmetric, but not conversely in either case. Some properties of these rings are discussed in the first two sections. In pseudo symmetric rings, the minimal prime ideals have the same characterization as for the commutative case. Mewborn's characterization of a commutative ring with compact minimal prime ideal space is generalized to pseudo symmetric rings with certain right quotient ring. For a pseudo symmetric ring, its prime ideal space is a T1-space iff it is a completely regular T2-space iff its usual basic open sets are closed as well. Presented to the Society, November 24, 1972 and January 28, 1973 under the title Prime ideal space and sheaf representation; received by the editors October 16, 1972. AMS (MOS) subject classifications (1970). Primary 16A64, 16A66, 18F20, 16A34; Secondary 16A48, 54D10, 54D20, 54H10.
222 citations
TL;DR: In this paper, it was shown that any nonzero ideal of a semiprime ring with polynomial identity intersects the center nontrivially (Theorem 2).
Abstract: Introduction. The purpose of this paper is to provide a fresh outlook to various questions on rings with polynomial identity by examining the centers of such rings. This approach yields the interesting result that any nonzero ideal of a semiprime ring with polynomial identity intersects the center nontrivially (Theorem 2). There are at least two interesting consequences to Theorem 2: a generalization of Wedderburn's theorem (any semiprimitive ring with polynomial identity, whose center is a field, is simple) and a strengthening of Posner's theorem [1] (any prime ring with a polynomial identity has a simple ring of quotients whose center is the quotient field of the center of the prime ring). The proofs are elementary modulo Jacobson [3]. Of course rings are not necessarily commutative and for the sake of simplicity we assume a unit 1. The key argument in this paper is an application of Formanek's central polynomials for matrix algebras over a field, whose important properties are [2] : Let Mn be an n x n matrix algebra over an arbitrary field. Then there exists a polynomial gn(Xl9.. .,Xm) which has coefficients in Z; is homogeneous (degree > 0) in every variable and linear in all but the first variable; takes values in the center for every specialization in Mn; and is nonvanishing for some specialization.
166 citations
TL;DR: In this article, a universal mode chart for the microstrip ring resonator based on a radial waveguide model is presented, where the resonant frequency is related to the width of the ring conductor.
Abstract: A universal mode chart for the microstrip ring resonator, based on a radial waveguide model, is presented. The resonant frequency is related to the width of the ring conductor. Experimental results from 4 to 16 GHz are shown to be in good agreement with the theory.
149 citations
TL;DR: Forster and Swan as mentioned in this paper used the notion of a basic element (in the terminology of Swan) to extend Bass's theorem to the case of finitely generated modules and showed that any torsion-free module over an integrally closed ring is an extension of a free module.
146 citations
Book•
01 Jan 1973
TL;DR: The closed components of R-Divisible and cotorsion modules as discussed by the authors are simple divisible modules, semi-simple and uniserial modules, the integral closure, the primary decomposition, and the first neighborhood ring.
Abstract: h-Divisible and cotorsion modules.- Completions.- Compatible extensions.- Localizations.- Artinian divisible modules.- Strongly unramified ring extensions.- The closed components of R.- Simple divisible modules.- Semi-simple and uniserial divisible modules.- The integral closure.- The primary decomposition.- The first neighborhood ring.- Gorenstein rings.- Multiplicities.- The canonical ideal of R.
134 citations
TL;DR: In this paper, a construction of irreducible unitary representations of the group of continuous transformations, where is a compact space with a measure and, that commute with transformations in preserving.
Abstract: We obtain a construction of the irreducible unitary representations of the group of continuous transformations , where is a compact space with a measure and , that commute with transformations in preserving .This construction is the starting point for a non-commutative theory of generalized functions (distributions). On the other hand, this approach makes it possible to treat the representations of the group of currents investigated by Streater, Araki, Parthasarathy, and Schmidt from a single point of view.
121 citations
TL;DR: The Göttingen State and University Library provides access to digitized documents strictly for noncommercial educational, research and private purposes and makes no warranty with regard to their use for other purposes.
Abstract: The Göttingen State and University Library provides access to digitized documents strictly for noncommercial educational, research and private purposes and makes no warranty with regard to their use for other purposes. Some of our collections are protected by copyright. Publication and/or broadcast in any form (including electronic) requires prior written permission from the Goettingen Stateand University Library. Each copy of any part of this document must contain there Terms and Conditions. With the usage of the library's online system to access or download a digitized document you accept there Terms and Conditions. Reproductions of material on the web site may not be made for or donated to other repositories, nor may be further reproduced without written permission from the Goettingen Stateand University Library For reproduction requests and permissions, please contact us. If citing materials, please give proper attribution of the source.
115 citations
TL;DR: In this article, a partial solution to the problem of characterizing the rings R for which Abelian groups can be characterized is presented, which is the same problem posed by Fuchs in his well-known book "Abelian Groups".
Abstract: One of the still unsolved problems posed by Fuchs in his well-known book “Abelian Groups” [2] is Problem 45: characterize the rings R for which . I present here a partial solution.
78 citations
TL;DR: In this paper, the dimensions of major ring features and the disk of the planet were discussed, and the observed photometric parameters of the ring were compared with those of various forms of snow.
73 citations
TL;DR: In this article, the persistent flow of a superfluid in a ring is discussed in terms analogous to those previously used for superconductors and the existence of a phase memory around the ring is shown to be responsible for energy minima with a periodic dependence on the total momentum which is directly related to the quantization of circulation.
Abstract: The persistent flow of a superfluid in a ring is discussed in terms analogous to those previously used for superconductors. The existence of a phase memory around the ring is shown to be responsible for energy minima with a periodic dependence on the total momentum which is directly related to the quantization of circulation. The general features are illustrated by means of the ideal Bose gas and the model of quasiparticles as examples.
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TL;DR: Weakly regular rings as discussed by the authors generalize a property of regular rings, namely, I2=I for every right (left) ideal, and they are called right(left) weakly regular.
Abstract: This paper attempts to generalize a property of regular rings, namely,I2=I for every right (left) ideal. Rings with this property are called right (left) weakly regular. A ring which is both left and right weakly regular is called weakly regular. Kovacs in [6] proved that, for commutative rings, weak regularity and regularity are equivalent conditions and left open the question whether for arbitrary rings the two conditions are equivalent. We show in §1 that, in general weak regularity does not imply regularity. In fact, the class of weakly regular rings strictly contains the class of regular rings as well as the class of biregular rings.
TL;DR: The invariant prime ideals in the complex bordism ring of finite complexes were derived in this paper. But their invariance was not proved for all stable stable $MU$-cohomology operations.
Abstract: We determine the prime ideals in the complex bordism ring $\pi_{\ast}(MU)$ which can be the annihilator ideals of elements in the complex bordism of finite complexes. Such a prime ideal must be finitely generated and invariant under all stable $MU$-cohomology operations. We go on to determine the invariant prime ideals in $\pi_{\ast}(MU)$.
TL;DR: In this article, the Galois-Eisenstein-Ore rings (GEO-rings) are defined, and the existence of a distinguished basis in a fintie bimodule over a Galois ring is proved.
Abstract: Every such ring is a direct sum of matrix rings over finite completely primary principal ideal rings. These latter rings are called Galois-Eisenstein-Ore rings or GEO-rings. A number of defining properties for GEO-rings are given, from which it follows that a finite ring with identity in which every two-sided ideal is left principal is a principal ideal ring. A theorem on the existence of a distinguished basis in a fintie bimodule over a Galois ring is proved, generalizing a similar theorem of Raghavendran. Finally, a GEO-ring is described as the quotient ring of an Ore polynomial ring over a Galois ring by an ideal of a special form, generated by Eisenstein polynomials. Bibliography: 10 items.
TL;DR: For a locally compact group G with property (PI), if there is a continuous projection of L1(G) onto a closed left ideal I, then there is an approximate identity in I.
Abstract: For a locally compact group G with property (PI), if there is a continuous projection of L1(G) onto a closed left ideal I, then there is a bounded right approximate identity in I. If I is further 2-sided, then I has a 2sided approximate identity. The converse is proved for w*-closed left ideals. Let G be further abelian and let I be a closed ideal in L 1(G). The condition that I has a bounded approximate identity is characterized in a number of ways which include (1) the factorability of I, (2) that the hull of I is in the discrete coset ring of the dual group, and (3) that I is the kernel of a closed element in the discrete coset ring of the dual group. Introduction. Let G be a locally compact group, I a closed left ideal in L l(G) and P a continuous projection of L 1(G) onto I. It is proved by W. Rudin [I1, Theorem 1] that, if G is compact, there exists a continuous projection Q of L1(G) onto I such that (*) / * Qg = Q (/ * g) (/, g E L1(G)). Further [1, Proof of Theorem 2], if in addition G is abelian, then there exists an idempotent measure p on G such that Qf = f* t (f cL (G)) so that Q is actually an algebra homomorphism. It follows that I, considered as a Banach algebra, has a bounded approximate identity. The purpose of Part I of this paper is to find out what happens if G is not compact or abelian. It turns out that if G has the property (P ) (which it does if it is compact) then the projection P leads to a net of projections Q for which the formula (*) "almost" holds, and that I still has a bounded (right) approximate identity (Theorem 2). Received by the editors December 10, 1971. AMS (MOS) subject classifications (1970). Primary 22B10, 22D15, 43A20; Secondary 43A45, 46H 10.
TL;DR: In this paper, the Cay ley -Hami l ton theorem was used to show that a 2 • 2-matr ix ring is an n • wi th f~+i = 0 t hen (T r f ) m"+l = 0.
Abstract: The origin of this paper is a mispr int (?) in Bourbak i ([4], p. 156, Exerc ise 13 d). There it is s t a ted t ha t i f f is a 2 • 2-matr ix wi th entries in a c o m m u t a t i v e ring and f 2 = 0 t hen (Tr f )4 = 0 and 4 is the smallest integer wi th this p roper ty . Using the Cay ley -Hami l ton theorem we get f2 _ af ~b l = 0 where a = T r f and b = d e t f . Not ing t h a t f 2 = 0 and taking t races we g e t a T r f = a 2 = 2b. Mult iplying the f irs t equa t ion by f gives bf = 0 which implies b 9 Tr f = ba = O. Hence a a = 2ab = 0 so 3 and not 4 is the smallest integer above. Expe r imen t ing wi th small m and n one soon makes the conjecture: I f f is an n • wi th f~+i = 0 t hen ( T r f ) m"+l = 0. This is p roved in a somewhat more general set t ing in 1.7 using ex te r ior algebra. I n Sect ion 1 the character is t ic polynomia l 2t(f) is def ined for an endomorph ism f : P ~ P where P is a f in i te ly genera ted projec t ive A-module (A is a commu ta t i ve ring wi th 1). I f P is free then 2t(f) = det (1 ~tf). The exponent ia l t race formula (in case A contains Q)
Book•
31 Dec 1973
TL;DR: In this article, the authors introduce ring and category theory and introduce commutative algebra, including the notion of projectives, injectives, flat modules, and homological dimensions.
Abstract: Introduction Part I. Introductory ring and category theory: General definitions, notations, example Basic properties of projectives, injectives, flat modules, Hom and $\otimes$ Basic commutative algebra Part II. Homological dimensions: Definitions of various dimensions, Ext, and Tor An alternative derivation of Tor and Ext Elementary applications Commutative algebra revisited Set theoretic propositions Not so elementary applications and counting theorems More counting Appendix. Introductory set theory Notation, definitions, basic axioms Cardinals, ordinals, and the axiom of choice Bibliographical notes.
01 Mar 1973
TL;DR: In this article, a generalization of the concept of a decomposition of a ring into a direct sum of ideals is introduced and the question of semisimplicity of the ring in terms of its summands is investigated.
Abstract: A generalization of the concept of a decomposition of a ring into a direct sum of ideals is introduced. The question of semisimplicity of the ring in terms of the semisimplicity of its summands is investigated. The results are applied to semigroup rings.
01 Feb 1973
TL;DR: In this article, it was proved that any commutative extension of a local Cohen-Macaulay local ring by a canonical module is a Gorenstein ring, which is a special case of the Hochschild extension.
Abstract: Reiten has demonstrated that the trivial Hochschild extension of a Cohen-Macaulay local ring by a canonical module is a Gorenstein local ring. Here it is proved that any commutative extension of a Cohen-Macaulay local ring by a canonical module is a Gorenstein ring. Also Gorenstein extensions of a local Cohen- Macaulay ring by a module are studied.
TL;DR: In this article, the authors show that if there is a unitary ring homomorphism R→S, then it is not difficult to show that cdsC ≦ cdRC.
Abstract: Let R be a ring with identity, and let C be a small, nonempty category. We denote the category of right R-modules by AbR and the category of contravariant functors C → AbR by AbRC*. The limit functor is left exact, and its kth right derived functor is denoted by colim k . The R-cohomological dimension of C is defined by If there is a unitary ring homomorphism R→S, then it is not difficult to show that cdsC ≦ cdRC.
01 Feb 1973
TL;DR: In this paper, it was shown that the theory of commutative regular rings without nilpotent elements has a model companion, which is decidable and is the model completion of the theory.
Abstract: We show that the theory of commutative rings without nilpotent elements has a model companion. The model companion is decidable and is the model completion of the theory of commutative regular rings. Recall that a theory K is model-complete if for any model M of K, KUD(M) is complete, where D(M) denotes the diagram of M. A natural generalization of this notion is that of a model completion. We say that K' is a model completion of K if K' extends K and, for any model M of K, K'UD(M) is consistent and complete (see [5]). For example the theory of algebraically closed fields is a model completion of the theory of fields and the theory of real closed fields is a model completion of the theory of ordered fields. A further generalization is the idea of a model companion. We say that K and K' are mutually model consistent if every model of K can be embedded in a model of K' and vice versa. K' is a model companion of K if K and K' are mutually model consistent and K' is model-complete. Model completions and model companions (when they exist) are unique. For this and other elementary properties see [5] and [6]. In everything that follows we shall use the word ring to mean ring with identity. We call a ring R regular (in the sense of von Neumann) if for any x e R there exists y e R such that xyx=x. (A good general reference for the algebra relevant to this paper is [3].) Notice that in any commutative ring the set of idempotents forms a Boolean algebra under the operations euf=e+f-ef, enf=ef. Hence when we say that e is a subidempotent off we mean that ef=e (i.e. enf=e). e is a minimal idempotent if ef=f implies thatf is either e or 0. We shall say that a quantifierfree formula ?p(a1, ... , an) holds on an idempotent e of a ring R if the formula obtained from ?p by multiplying every term in ?p by e holds in R. Received by the editors February 25, 1972 and, in revised form, July 28, 1972. AMS (MOS) subject classifications (1970). Primary 02H05, 02H99; Secondary 02G05, 02G20.
TL;DR: In this paper, the authors studied the structure of the maximal quotient ring of a nonsingular ring R by looking at its prime ideals and showed that the primes of R separate into two types, called essential and closed, and that for any prime P, the two-sided ideals in the ring R/P are linearly ordered.
Abstract: Although the notion of the maximal quotient ring of a nonsingular ring has been around for some time, not much is known about its structure in general beyond the important theorems of Johnson and Utumi [4; 11] that it is von Neumann regular and self-injective. The purpose of this paper is to study the structure of such a regular, self-injective ring R by looking at its prime ideals. Initially, we show that the primes of R separate into two types, called ‘'essential” and ‘“closed”, and that for any prime P, the two-sided ideals in the ring R/P are linearly ordered.
TL;DR: The finite generation of the ring of invariants of a special class of unipotent groups is established-namely, unIPotent radicals of parabolic subgroups.
Abstract: The finite generation of the ring of invariants of a special class of unipotent groups is established—namely, unipotent radicals of parabolic subgroups
TL;DR: In this article, it was shown that if G is a torsion-free group and F is a field, then the group ring F[G] is a ring without zero divisors.
Abstract: If G is a torsion–free group and F is a field, is the group ring F[G] a ring without zero divisors? This is true if G is an ordered group or various generalizations thereof - beyond this the question remains untouched. This paper proves a related result.
TL;DR: In this paper, it has been shown that a map of the euclidean plane, T: E2 E2, which preserves unit distance is an isometry, using an approximation procedure based on the theorem of Kronecker that the ring generated over Z by J3 is dense in R.
Abstract: It has been proved by P. Zvengrowski [1, appendix to chapter II] that a map of the euclidean plane, T: E2 E2, which preserves unit distance is an isometry. His proof uses an approximation procedure based on the theorem of Kronecker that the ring generated over Z by J3 is dense in R . A weaker version of this theorem, where the map is assumed to be one-to-one, is proved in [2]. Some of the unproved assertions in [2] are not so obvious as one might think, since the proofs we have, anyway, do not generalize to higher dimensions. However, the ideas in [2] could be used to considerably simplify Zvengrowski's proof, eliminating the reliance on algebraic number theory. Here we present elementary proofs for all dimensions of this theorem.
TL;DR: In this article, it was shown that the first cohomology groups are isomorphic to certain subgroups of the group of units in the ring modulo p-th powers.
Abstract: The intent of this paper is to determine the first flat cohomology groups of certain finite fiat group schemes which are defined over the spectrum of the ring of integers in a local number field. We discover that the first cohomology groups are isomorphic to certain subgroups of the group of units in the ring modulo p-th powers. Our main result, Theorem 1, was announced in [M-R, Prop. 9.3]. I would like to express my thanks to Professor Barry Mazur for his generous interest and encouragement in this work. Throughout we will consistently use the following notation: K is a local number field with ring of integers R; U is the group of units in R, ord is the additive valuation which takes R surjectively to Z; U(M { u C U: ord (1 - u) ? i}, the residue field k of R is assumed to have characteristic p, and we shall regard P. = Z/pZ as being a subfield of k; the number of elements in kc is q =- pf; e = e(K/Qp) will denote the absolute ramification index of K over Q,. We will always assume that K contains the p-th roots of unity; among other things this implies that -p is a p - 1-st power in R and that m = e/ (p -11 is an integer. Ks will denote a fixed separable closure of K. All our group schemes will be flat over Spec (R) and will be considered as inducing sheaves for the (fppf)- or (fpqf)-site over Spec(R) [SGA 3, IV 6.3].
01 Feb 1973
TL;DR: In this paper, it was shown that the similarity of certain invertible matrices over a finite ring can be reduced to questions of similarity over finite fields through the application of canonical epimorphisms.
Abstract: It is shown that questions of similarity of certain invertible matrices over a finite ring can be reduced to questions of similarity over finite fields through the application of canonical epimorphisms. Suprunenko has shown in [3] that two invertible matrices over ZIZm whose orders are relatively prime to m are similar if and only if their canonical images are similar over the fields Z/Zp for each prime divisor p of m. An analogous result holds for invertible matrices over any finite commutative ring with identity. Preliminaries. If R is a finite commutative ring with identity, then R is uniquely a ring direct product of finite local rings [1, Theorem 8.7, p. 90]. Suppose that R=]7Lf Ri, where Ri is a finite local ring with maximal ideal Mi. Each Ri has cardinality pzi for some prime p and has associated with it a canonical projection, hi: Ri -RiMi = GF(p ). Setting ki=GF(pfi) we will say that the finite fields {ki:i=l, 2, , t} are the fields associated with R. Observe that the decomposition of R carries over to the general linear group of degree n over R yielding GLn(R)FJ7LJ1GL,(RJ). Furthermore, for each i, the projection hi induces an epimorphism, hi: GL (Ri) -GLn(ki). If GLn(R,) is taken as the group of n by n invertible matrices over Ri, then hi is simply reduction modulo Mi. Note that the kernel of hi, Ki, has cardinality a power of pi and thus is a solvable group. The following corollary to P. Hall's extension of the Sylow theorems [2, Theorem 9.3.1, p. 141] is the key result needed for Theorems 1 and 2. Observation. Let G be a finite group with solvable normal subgroup K and let C=G/K={glg E G}. Let o and f, be elements of G with (IoI, IkI)= 1=(1II, IKI). Then o-j implies oc-z . Received by the editors April 12, 1972. AMS (MOS) subject classifications (1970). Primary 13H99, 15A33, 15A21, 20D20, 20H25.
01 Mar 1973
TL;DR: In this article, it was shown that a semilattice is regular if and only if each semigroup ring is regular, in the sense of von Neumann (6), if for each a e R there exists x e R such that axa=a.
Abstract: If R is a supplementary semilattice sum of subrings Ra, a e Cl, then R is regular if and only if each Rat is regular. A ring is said to be regular, in the sense of von Neumann (6), if for each a e R there exists x e R such that axa=a. The concept of (supplementary) semilattice sum is defined in the previous article (8). In this article, we prove that if R is a supplementary semilattice sum of subrings Rx, a e £2, then R is regular if and only if Rx is regular for every a e £2. We state, without proof, an application of this result to the regularity of semigroup rings. Throughout this paper D will denote a semigroup. Definitions of any concepts not defined herein will be found in (1) or (8). We first prove the main theorem in the case when the semilattice has