Essentially Distinguished Modules

Let R be a commutative ring with identity and M be a unitary (left) R-module. In this paper we present the concepts essentially distinguished module and essentially distinguished ring as generalizations of distinguished module and distinguished ring. M is called essentially distinguished R-module provided that ?ann?_M ( I ) is an essential sub module of M for each maximal ideal I of R. R is called R-distinguished ring if R as an R-module is distinguished. The basic properties of such modules (rings) are studied.