Modules Over Matrix Near-Rings and the ...-Radical.
We study the class of modules which are invariant under idempotents of their envelopes. We say that a module M is -idempotent-invariant if there exists an -envelope u : M → X such that for any idempotent g ∈ End(X) there exists an endomorphism f : M → M such that uf = gu. The properties of this class of modules are discussed. We prove that M is -idempotent-invariant if and only if for every decomposition , we have . Moreover, some generalizations of -idempotent-invariant modules are considered....
We characterize the semiregularity of the endomorphism ring of a module with respect to the ideal of endomorphisms with large kernel, and show some new classes of modules with semiregular endomorphism rings.
The present work gives some characterizations of -modules with the direct summand sum property (in short DSSP), that is of those -modules for which the sum of any two direct summands, so the submodule generated by their union, is a direct summand, too. General results and results concerning certain classes of -modules (injective or projective) with this property, over several rings, are presented.