It is proved that every commutative ring whose RD-injective modules are Σ-RD-injective is the product of a pure semi-simple ring and a finite ring. A complete characterization of commutative rings for which each artinian (respectively simple) module is RD-injective, is given. These results can be obtained by using the properties of RD-flat modules and RD-coflat modules which are respectively th...

We construct analogues of FI-modules where the role of the symmetric group is played by the general linear groups and the symplectic groups over finite rings and prove basic structural properties such as Noetherianity. Applications include a proof of the Lannes– Schwartz Artinian conjecture in the generic representation theory of finite fields, very general homological stability theorems with t...

Let R be a commutative ring. An R-module M is called co-multiplication provided that foreach submodule N of M there exists an ideal I of R such that N = (0 : I). In this paper weshow that co-multiplication modules are a generalization of strongly duo modules. Uniserialmodules of finite length and hence valuation Artinian rings are some distinguished classes ofco-multiplication rings. In additio...

The aim of this paper is to investigate generalizations locally artinian supplemented modules in module theory, namely radical and strongly modules. We have obtained elementary features for them. Also, we characterized by left perfect rings. Finally, proved that the reduced part a $R$-module has same property over Dedekind domain $R$.

There exists an analogy between the structure of ideals of a cone in a right-ordered group and the structure of ideals of a Dubrovin valuation ring in a simple artinian ring. The structure of ideals of rank one cones and rank one Dubrovin valuation rings can be described completely. One sided versions of this problem are considered in 5] where the ideal theory of right cones is developed and a ...