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 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 some Young modules

Let \({\Sigma_r}\) be the symmetric group acting on \({r}\) letters, \({K}\) be a field of characteristic 2, and \({\lambda}\) and \({\mu}\) be partitions of \({r}\) in at most two parts. Denote the permutation module corresponding to the Young subgroup \({\Sigma_\lambda}\), in \({\Sigma_r}\), by \({M^\lambda}\), and the indecomposable Young module by \({Y^\mu}\). We give an explicit presentation of the endomorphism algebra \({{\rm End}_{k[\Sigma_r]}(Y^\mu)}\) using the idempotents found by Doty et al. (J Algebra 307(1):377–396, 2007).     

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 of essential extensions of a noetherian module

Nil subrings of the ring of endomorphisms of the rational completion of a noetherian module are nilpotent. If the quasi-injective hull of a noetherian module is contained in its rational completion, then the ring of endomorphisms of the former is semi-primary.