On generalizations of ADS modules and rings

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.