Primary Submodules over a Multiplicatively Closed Subset of a Commutative Ring

In this paper, we introduce the concept of primary submodules over S which is a generalization of the concept of S-prime submodules. Suppose S is a multiplicatively closed subset of a commutative 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 ∈ S such that, for all a ∈ R, m ∈ M, am ∈ Q implies that sm ∈ Q or san ∈ (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 ∈ S.