General distributivity and thickness of modules

Abstract EN

Let ω > 2 be a cardinal and S a class of semisimple left R-modules (closed under isomorphisms).

A module RM is called w-thick relative to S, if dim S < w for each subfactor S of M with S Є S.

This notion allows to study from a unified point of view w-distributive modules, i.e., modules satisfying A +\λ∈ΛBλ=\λ∈Λ(A +\µ∈Λ\{λ}Bµ)for all submodules A and families {Bλ}A of submodules with cardinality |A| = ω, and ω -thick modules, i.e., modules which are w-thick relative to the class of all semisimple left R-modules.

In particular 2-distributive modules coincide with distributive modules, 2-thick modules coincide with uniserial modules, N0-thick mod¬ules coincide with q.f.d.

modules, i.e., modules whose factor modules have finite uniform dimension.

We also consider relative w-quasi-invariant, relative ω-noetherian and relative ω-Bezout modules.

Prop¬erties of modules from these classes are investigated including the relationship between them.

Moreover, for modules RM and RU, the relationship between w-distributivity of M and properties of the left EndR(U)- module HomR(U,M) and the right EndR(U)-module HomR(M, U) are studied.