On generalizations of ADS modules and rings

Abstract

A rightmodule M over a ring R is said to be ADSif, for every decomposition M = S ⊕ T and every complement T' of S, we have M = S ⊕ T'. In this article, we study and provide several new characterizations of new class of essential modules and generalization of ADS modules. We prove that M is semisimple if and only if every module in σ[M] is generalized ADS if and only if every generalized ADS module in σ[M] is M-injective.