Primary Submodules over a Multiplicatively Closed Subset of a Commutative Ring

In this paper, we introduce the concept of primary sub- modules over S which is a generalization of the concept of S-prime submodules. Suppose S is a multiplicatively closed subset of a commu- tative ring R and let M be a unital R-module. A proper submodule Q of M with (Q :R M) / S = ; is called primary over S if there is an s 2 S such that, for all a 2 R, m 2 M, am 2 Q implies that sm 2 Q or san 2 (Q :R M), for some positive integer n. We get some new results on primary submodules over S. Furtheremore, we compare the concept of primary submodules over S with primary ones. In particular, we show that a submodule Q is primary over S if and only if (Q :M s) is primary, for some s 2 S.