Essential Submodules with respect to an Arbitrary Submodule

The concept of essential submodules is a well known concept. In this paper we try to replace an arbitrary submodule of M, say T, instead of 0 in the definition of essential submodules. By this, essential submodules are precisely {0}-essential submodules. For a submodule K of right R-module M, we have K ess M if and only if (K : m) is annr(m)-essential right ideal of R, for each m 2 M \ {0}. Among other things, this generalization of essential submodules gives a necessary and sufficient condition for M T being finitely co-generated.