On Weakly S-primary Submodules

Abstract

Let R be a commutative ring with a non-zero identity, S be a mul-
tiplicatively closed subset of R and M be a unital R-module. In this paper, we
define a submodule N of M with (N :R M) \ S = ; to be weakly S-primary
if there exists s 2 S such that whenever a 2 R and m 2 M with 0 6= am 2 N,
then either sa 2
p
(N :R M) or sm 2 N. We present various properties and
characterizations of this concept (especially in finitely generated faithful mul-
tiplication modules). Moreover, the behavior of this structure under module
homomorphisms, localizations, quotient modules, cartesian product and ide-
alizations is investigated. Finally, we determine some conditions under which
two kinds of submodules of the amalgamation module along an ideal are weakly
S-primary.