Endomorphism rings of completely pure-injective modules

Let be a ring, its injective envelope, and the Jacobson radical of . It is shown that if every finitely generated submodule of embeds in a finitely presented module of projective dimension , then every finitley generated right -module is canonically isomorphic to . This fact, together with a well-known theorem of Osofsky, allows us to prove that if, moreover, is completely pure-injective (a property that holds, for example, when the right pure global dimension of is and hence when is a countable ring), then is semiperfect and is finite-dimensional. We obtain several applications and a characterization of right hereditary right noetherian rings.

     

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.

     

Endomorphism rings of modules and lattices of submodules

This survey covers material inReferativnyi Zhurnal Matematika and is a continuation of A. V. Mikhalev's survey in Algebra. Topology. Geometry, Vol. 12 (Itogi Nauki i Tekhniki (1974)). Papers on endomorphism rings of modules over associative rings and on lattices of submodules are considered.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 21, pp. 183–254, 1983.     

Endomorphism rings and the category of finitely generated projective modules

Given a left R-module M, we study the connection between the (right and left) properties of its endomorphism ring S=End(_RM) and the properties of the category σ _f [M] of all submodules of finitely M-generated left R-modules.     

Endomorphism rings of bimodules

Let \(M\) be an \(R\) - \(R\) -bimodule over a semi-prime right and left Goldie ring \(R\) . We investigate how non-singularity conditions on \(M_R\) are related to such conditions on \(_RM\) . In particular, we say an \(R\) - \(R\) -bimodule \(M\) such that \(_RM\) and \(M_R\) are non-singular has the right essentiality property if \(IM_R\) is essential in \(M_R\) for all essential right ideals \(I\) of \(R\) , and investigate several questions related to this property.     

On some classes of modules and characterization of rings

An abstract class of R-Mod is termed Additive Class if it is closed under taking submodules, homomorphic images and finite direct sums. In this paper we study the lattice properties of R-ad, the big lattice of all additive classes in R-mod for R any ring. Also, we study the lattice properties of R-ad when R is a Noetherian or semisimple ring. We describe a characterization of some rings through the additive classes and their properties. We study some properties of additive classes related with the associated linear filters.