On the equivalence of codes over rings and modules

In light of the result by Wood that codes over every finite Frobenius ring satisfy a version of the MacWilliams equivalence theorem, a proof for the converse is considered. A strategy is proposed that would reduce the question to problems dealing only with matrices over finite fields. Using this strategy, it is shown, among other things, that any left MacWilliams basic ring is Frobenius. The results and techniques in the paper also apply to related problems dealing with codes over modules.     

Elementary equivalence of categories of modules over rings, endomorphism rings, and automorphism groups of modules

In this paper we give a brief review of some recent results of elementary equivalence of linear and algebraic groups and our last new results of elementary equivalence of categories of modules, endomorphism rings of modules, lattices of submodules of modules, and automorphism groups of modules. __________ Translated from Fundamentalnaya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 10, No. 2, pp. 51–134, 2004.     

Endomorphism rings of simple modules over group rings

If is a finitely generated nilpotent group which is not abelian-by-finite, a field, and a finite dimensional separable division algebra over , then there exists a simple module for the group ring with endomorphism ring . An example is given to show that this cannot be extended to polycyclic groups.

     

Stable equivalence and rings whose modules are a direct sum of finitely generated modules

The representation theory of a ring Δ has been studied by examining the category of contravariant (additive) functors from the category of finitely generated left Δ-modules to the category of abelian groups. Closely connected with the representation theory of a ring is the study of stable equivalence, which is a relaxing of the notion of Morita equivalence. Here we relate two stably equivalent rings via their respective functor categories and examine left artinian rings with the property that every left Δ-module is a direct sum of finitely generated modules.     

Finite modules over non-semisimple group rings

LetG be a fixed graph and letX _G be the number of copies ofG contained in the random graphG(n, p). We prove exponential bounds on the upper tail ofX _G which are best possible up to a logarithmic factor in the exponent. Our argument relies on an extension of Alon’s result about the maximum number of copies ofG in a graph with a given number of edges. Similar bounds are proved for the random graphG(n, M) too. Research of the second author supported by KBN grant 2 P03A 027 22. Research of the third author supported by KBN grant 2 P03A 15 23.     

Quadratic modules and the orthogonal group over polynomial rings

The following conjecture of Parimala is proved: Any quadratic space over a polynomial ring with coefficients from an algebraically closed field of characteristic different from 2 is extended from the coefficient field. In the case of an arbitrary field of characteristic different from 2, an analogous result is obtained for quadratic spaces whose Witt index is at least 2. Also proved are general cancellation theorems for quadratic modules and a stabilization theorem for the orthogonal group over arbitrary polynomial rings.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta, im. V. A. Steklova AN SSSR, Vol. 71, pp. 216–250, 1977.     

On Schur classes for modules over group rings

We consider the problem of coupling between a quotient module A/C _A (G) and a submodule A(ωRG), where G is a group, R is a ring, and A is an RG-module; C _A (G) can be considered as an analog of the center of the group, and the submodule A(ωRG) can be considered as an analog of the derived subgroup of the group. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 9, pp. 1261–1268, September, 2007.     

On direct decompositions in modules over group rings

In the theory of infinite groups, one of the most important useful generalizations of the classical Maschke theorem is the Kovačs-Newman theorem, which establishes sufficient conditions for the existence of G-invariant complements in modules over a periodic group G finite over the center. We genralize the Kovačs-Newman theorem to the case of modules over a group ring KG, where K is a Dedekind domain. Dnepropetrovsk University, Dnepropetrovsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 255–261, February, 1997.     

Note on inertia properties of filtrations on group graded rings and modules

The idea of a control subgroup or inertia group for objects over group graded rings is applied to filtratons of the objects defined over the graded ring. we stidy the interplay between properties of a foltered nature and induction from subgroups of the grading groups.     

Bezout modules and rings

For any ring A, there exist a Bezout ring R and an idempotent e ∈ R with A ? eRe. Every module over any ring is a direct summand of an endo-Bezout module. Over any ring, every free module of infinite rank is an endo-Bezout module.