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.     

Elementary Equivalence of Categories of Modules and other Algebraic Structures

This paper is a review of some recent results of elementary equivalence of linear and algebraic groups and our recent new results of elementary equivalence of categories of modules, endomorphism rings of modules, and automorphism groups of modules. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 13, Algebra, 2004.     

Foxby equivalence, complete modules, and torsion modules

Taking the idea from classical Foxby equivalence, we develop an equivalence theory for derived categories over differential graded algebras. Both classical Foxby equivalence and the Morita equivalence for complete modules and torsion modules developed by Dwyer and Greenlees arise as special cases.     

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.     

On distributive modules and rings

In this paper we study ω-distributive modules, where ω is a cardinal number. We extend a characterization of distributive modules to ω-distributive modules. In particular, the case in which ω = n is a finite cardinal is considered. We apply the results to the case n = 2, obtaining new characterizations for distributive modules and rings. Special attention is given to saturated submodules and ideals.     

Second representable modules over commutative rings*

Let R be a commutative ring. We investigate R-modules which can be written as finite sums of second R-submodules (we call them second representable). The class of second representable modules lies between the class of finitely generated semisimple modules and the class of representable modules; moreover, we give examples to show that these inclusions are strict even for Abelian groups. We provide sufficient conditions for an R-module M to be have a (minimal) second presentation, in particular within the class of lifting modules. Moreover, we investigate the class of (main) second attached prime ideals related to a module with such a presentation.     

n-gr-coherent rings and Gorenstein graded modules

Let R be a graded ring and n ? 1 be an integer. We introduce and study the notions of Gorenstein n-FP-gr-injective and Gorenstein n-gr-flat modules by using the notion of special finitely presented graded modules. On n-gr-coherent rings, we investigate the relationships between Gorenstein n-FP-gr-injective and Gorenstein n-gr-flat modules. Among other results, we prove that any graded module in R-gr (or gr-R) admits a Gorenstein n-FP-gr-injective (or Gorenstein n-gr-flat) cover and preenvelope, respectively.

     

Differential projective modules over differential rings

Differential modules over a commutative differential ring which are projective as ring modules, with differential homomorphisms, form an additive category. Every projective ring module is shown occurs as the underlying module of a differential module. Differential modules, projective as ring modules, are shown to be direct summands of differential modules free as ring modules; those which are differential direct summands of differential direct sums of the ring being induced from the subring of constants. Every differential module finitely generated and projective as a ring module is shown to have this form after a faithfully flat finitely presented differential extension of the base.     

Characterization of rings using quasiprojective modules

Semisimple, semiperfect, and perfect rings are characterized by quasiprojective modules and quasiprojective covers over them. The results in this note are taken from the author’s doctoral dissertation, being written at the Hebrew University of Jerusalem under the direction of Professor S.A. Amitsur.     

On modules with the Kulikov property and pure semisimple modules and rings

For an R-module M let σ[M] denote the category of submodules of M-generated modules. M has the Kulikov property if submodules of pure projective modules in σ[M] are pure projective. The following is proved: Assume M is a locally noetherian module with the Kulikov property and there are only finitely many simple modules in σ[M]. Then, for every n ε , there are only finitely many indecomposable modules of length n in σ[M].

With our techniques we provide simple proofs for some results on left pure semisimple rings obtained by Prest and Zimmermann-Huisgen and Zimmermann with different methods.     

Local rings whose modules are almost serial

The present paper is a sequel to our previous work on almost uniserial rings and modules, which appeared in the Journal of Algebra in 2016; it studies rings over which every (left and right) module is almost serial. A module is almost uniserial if any two of its submodules are either comparable in inclusion or isomorphic. And a module is almost serial if it is a direct sum of almost uniserial modules. The results of the paper are inspired by a characterization of Artinian serial rings as rings having all left (or right) modules serial. We prove that if R is a local ring and all left R-modules are almost serial then R is an Artinian ring which is uniserial either on the left or on the right. We also produce a connection between local rings having all left and right modules almost serial, local balanced rings studied by Dlab and Ringel and local Köthe rings. Finally we prove Morita invariance of the almost serial property and list some consequences.     

Flat modules and rings finitely generated as modules over their center

A module is called distributive (is said to be a chain module) if the lattice of all its submodules is distributive (is a chain). Let a ringA be a finitely generated module over its unitary central subringR. We prove the equivalence of the following conditions:

1. A is a right or left distributive semiprime ring;
2. for any maximal idealM of a subringR central inA, the ring of quotientsA _M is a finite direct product of semihereditary Bézout domains whose quotient rings by the Jacobson radicals are finite direct products of skew fields;
3. all right ideals and all left ideals of the ringA are flat (right and left) modules over the ringA, andA is a distributive ring, without nonzero nilpotent elements, all of whose quotient rings by prime ideals are semihereditary orders in skew fields.

     

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.     

On semilocal modules and rings

It is well-known that a ring Ris semiperfect if and only if _RR (orR_R ) is a supplemented module. Considering weak supplementsinstead of supplements we show that weakly supplemented modules Mare semilocal (i.e.M/Rad(M) is semisimple) and that R is a semilocal ring if and only if _RR (orR_R ) is weakly supplemented. In this context the notion of finite hollow dimension (or finite dual Goldie dimension) of modules is of interest and yields a natural interpretation of the Camps-Dicks characterization of semilocal rings. Finitely generated modules are weakly supplemented if and only if they have finite hollow dimension (or are semilocal).     

Some new dimensions of modules and rings

We say that a class 𝒫 of right modules over a fixed ring R is an epic class if it is closed under homomorphic images. For an arbitrary epic class 𝒫, we define a 𝒫-dimension of modules that measures how far modules are from the modules in the class 𝒫. For an epic class 𝒫 consisting of indecomposable modules, first we characterize rings whose modules have 𝒫-dimension. In fact, we show that every right R-module has 𝒫-dimension if and only if R is a semisimple Artinan ring. Then we study fully Hopfian modules with 𝒫-dimension. In particular, we show that a commutative ring R with 𝒫-dimension (resp. finite 𝒫-dimension) is either local or Noetherian (resp. Artinian). Finally, we show that Mat_m(R) is a right Köthe ring for some m if and only if every (left) right module is a direct sum of modules of 𝒫-dimension at most n for some n, if and only if R is a pure semisimple ring.     

Elementary equivalence of rings with finitely generated additive groups

We give algebraic characterizations of elementary equivalence between rings with finitely generated additive groups. They are similar to those previously obtained for finitely generated nilpotent groups. Here, the rings are not supposed associative, commutative or unitary.     

Weakly regular modules over normal rings

Under study are some conditions for the weakly regular modules to be closed under direct sums and the rings over which all modules are weakly regular. For an arbitrary right R-module M, we prove that every module in the category σ(M) is weakly regular if and only if each module in σ(M) is either semisimple or contains a nonzero M-injective submodule. We describe the normal rings over which all modules are weakly regular.     

McCoy modules and related modules over commutative rings

Let M be a left R-module. Then M is a McCoy (resp., dual McCoy) module if for nonzero f(X)∈R[X] and m(X)∈M[X], f(X)m(X) = 0 implies there exists a nonzero r∈R (resp., m∈M) with rm(X) = 0 (resp., f(X)m = 0). We show that for R commutative every R-module is dual McCoy, but give an example of a non-McCoy module. A number of other results concerning (dual) McCoy modules as well as arithmetical, Gaussian, and Armendariz modules are given.     

Vertices, sources and Green correspondents of the simple modules for the large Mathieu groups

We investigate the simple modules for the sporadic simple Mathieu groups M₂₂, M₂₃ and M₂₄ as well as those of the automorphism group, the covering groups and the bicyclic extensions of M₂₂ in characteristics 2 and 3. We determine the vertices and sources as well as the Green correspondents of these simple modules. We also find two 3-blocks with elementary abelian defect groups of order 9 in these groups which are Morita equivalent to their Brauer correspondents.