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 submodulek 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 anecessary and sufficient condition for mt being finitely co-generated.