(m,n)-Closed submodules of modules over commutative rings

Abstract

Let R be a commutative ring and m, n be positive integers. We define a proper submodule N of an R-module M to be (m,n)-closed if for r is an element of R and b is an element of M, r(m)b is an element of N implies r(n) is an element of (N : M-R) or b is an element of N. This class of submodules lies properly between the classes of prime and primary submodules. Many characterizations, properties and supporting examples concerning this class of submodules are provided. The notion of (m,n) -modules is introduced and characterized. Furthermore, the (m,n)-closed avoidance theorem is proved. Finally, the (m,n)-closed submodules in amalgamated modules are studied.