Showing papers on "Torsion (algebra) published in 1979"

Torsion algebraic cycles and a theorem of Roitman

Abstract: © Foundation Compositio Mathematica, 1979, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Torsion-free modules over primary rings

Abstract: The Noetherian primary rings over which all indecomposable torsion-free modules in the sense of Bass have discrete normed endomorphism rings are described.

Some properties of two-dimensional surfaces with zero normal torsion in $e^4$

Abstract: In four-dimensional Euclidean space the author considers two-dimensional class surfaces having zero normal torsion at every point and in every direction. A necessary and sufficient condition is established for such surfaces to belong to a certain hyperplane. Bibliography: 3 titles.

The conormal module of an almost complete intersection

Abstract: ABsTRAcr. The conormal module of an ideal I in a commutative ring S is the S/I-module I/12. Assume S is a regular noetherian ring and I a prime ideal, which is locally everywhere a complete intersection or an almost complete intersection (i.e. needs one generator more than in the complete intersection case). In this situation necessary and sufficient conditions for 1/12 being torsion free are given. Moreover the torsion of I/12 is expressed in terms of Kihler differentials of S/I.

Some relations in the mod 3 cohomology ofH-spaces

Abstract: A third order non stable operation is constructed. Its domain and range are a linear subspace and a quotient space of the mod 3 cohomology but it represents a non linear transformation. Using this operation some relations over the Steenrod algebra in the module of primitives in the cohomology of anH-space are derived even in the 3 torsion free case.

Strongly prime group rings

Abstract: A ring R is (right) strongly prime (SP) if every nonzero two sided ideal contains a finite set whose right annihilator is zero, SP rings have been studied by Handelman and Lawrence who raised the problem of characterizing SP group algebras. They showed that if R is SP and G is torsion free Abelian, then the group ring RG is SP. The aim of this note is to determine some more group rings which are SP. For a ring R we also define the strongly prime radical s(R). We then show that s(R)G = s(W) for certain classes of groups.

The quotient field as a torsion-free covering module

Abstract: R will denote a commutative integral domain with quotient fieldQ. A torsion-free cover of a moduleM is a torsion-free moduleF and anR-epimorphism σ:F→M such that given any torsion-free moduleG and λ∈Hom R (G, M) there exists μ∈Hom R (G,F) such that σμ=λ. It is known that ifM is a maximal ideal ofR, R→R/M is a torsion-free cover if and only ifR is a maximal valuation ring. LetE denote the injective hull ofR/M thenR→R/M extends to a homomorphismQ→E. We give necessary and sufficient conditions forQ→E to be a torsion-free cover.

Morita equivalence of simple Noetherian rings

Abstract: We show that a simple Noetherian ring of finite global dimension and Krull dimension one is Morita equivalent to a domain. In [4] is given an example of a simple Noetherian ring that is not Morita equivalent to a domain. Now the ring in question has infinite global dimension but Krull dimension one. In contrast, we show here that if R is a simple Noetherian ring of finite global dimension and Krull dimension one, then R is Morita equivalent to a domain. This is a corollary of Theorem 1, which gives a generalisation of this result to simple rings of arbitrary finite Krull dimension. Throughout this note all rings will contain an identity and all modules will be unitary. We will denote the uniform dimension of a module M by ud(M) and the projective dimension by pd(M). The left Krull dimension of a ring R will be denoted by l-Kdim(R). THEOREM 1. Let R be a simple Noetherian ring of finite global dimension. Then R is Morita equivalent to a ring S such that ud(S) n + 1. Observe that, if P is an indecomposable finitely generated projective right S-module, then r 1 and a divides r. Further, any finitely generated projective right S-module P has ud(P) = ma for some integer m. The argument now closely follows that of [5, Lemma 2.8]. Let t be an integer > 1. Suppose, given any finitely generated torsion-free right S-module K with pd(K) S(m) M -O, for some integer m and module K. Now pd(K) = t 1 and so ud(K) = ca for some integer c. But, by [2, Lemma 3.6], uniform dimension is additive in this situation. So ud(M) = da for some integer d. Hence by induction and the fact that S has finite global dimension, every torsion free S-module has uniform dimension some integer multiple of a. But this implies that a = 1, a contradiction. COROLLARY 2. Let R be a simple Noetherian ring with finite global dimension and l-Kdim(R) = n 2n, then R contains nontrivial idempotents. PROOF. By the theorem, R is Morita equivalent to a ring S with ud(S) ud(S) + n. So, by [3, Theorem 5.2], P has a free direct summand. Hence R = I ED J for some nonzero right ideals I and J. Thus R has nontrivial idempotents. COROLLARY 3. Let R be a simple Noetherian ring with finite global dimension and l-Kdim(R) = 1. Then R is Morita equivalent to a domain. If R is not a domain, then R has nontrivial idempotents. Note that, as is shown by the example in [4] and [6], all three results are false if R has infinite global dimension. For rings of finite global dimension these corollaries give partial positive answers to [1, Questions 3 and 5, p. 381]. This content downloaded from 207.46.13.164 on Sun, 26 Jun 2016 06:12:40 UTC All use subject to http://about.jstor.org/terms

The additive groups of subdirectly irreducible rings II

Abstract: An abelian group G is said to be subdirectly irreducible if there exists a subdirectly irreducible ring R with additive group G . If G is subdirectly irreducible, and if every ring R with additive group G , and R 2 ≠ 0, is subdirectly irreducible, then G is said to be strongly subdirectly irreducible. The torsion, and torsion free, subdirectly irreducible and strongly subdirectly irreducible groups are classified completely. Results are also obtained concerning mixed subdirectly irreducible and strongly subdirectly irreducible groups.

Reidemeister torsion and group invariants of three-dimensional manifolds

Abstract: Two formulas are presented in this note. The first is purely algebraic and expresses the first elementary ideal of a finitely generated group in terms of the module of elementary derivatives. The second formula expresses the module of elementary derivatives of the fundamental group of a connected compact three-dimensionalpl -manifold with zero Euler characteristic in terms of the Reidemeister torsion of this manifold.

Modules over the cyclic group of prime order

Abstract: We describe the structure of all finitely generated modules over the integral group ring ZG, G= cyclic of prime order p. The additive groups of the modules in question need not be torsion free. We give a moderately detailed description of the indecomposable ZG-modules, and determine when two direct sums of such modules are isomorphic to each other.