PRIMARY SUBMODULES OVER A MULTIPLICATIVELY CLOSED SUBSET OF A COMMUTATIVE RING

In this paper, we introduce the concept of primary submodules overS which is a generalization of the concept of S-prime submodules. Suppose S isa multiplicatively closed subset of a commutative ring R and let M be a unitalR-module. A proper submodule Q of M with (Q :R M) \ S = ; is called primaryover S if there is an s 2 S such that, for all a 2 R, m 2 M, am 2 Q implies thatsm 2 Q or san 2 (Q :R M), for some positive integer n. We get some new resultson primary submodules over S. Furtheremore, we compare the concept of primarysubmodules over S with primary ones. In particular, we show that a submoduleQ is primary over S if and only if (Q :M s) is primary, for some s 2 S.