On weakly δR,M-semiprimary submodules

Abstract

Let R be a commutative ring with 1 not equal 0 and M be a nonzero unital R-module. Recall that a proper submodule N of M is called a semiprimary submodule of M if whenever rm is an element of N for r is an element of R and m is an element of M, then r is an element of root N : M or m is an element of root N, where rad(N) = root N is the intersection of all prime submodules of M containing N. We define a proper submodule N of M to be a weakly semiprimary submodule if whenever 0(M) not equal rm is an element of N for r is an element of R and m is an element of M, then r is an element of root N : M or m is an element of root N. In this paper, we give a characterization of generalizations of (weakly) semiprimary submodules. Let S(M) be the set of all submodules of M and delta(M) : S(M) -> S(M) be a function. Then we say delta(M) is an expansion of submodules of M if whenever A, B, C are submodules of M with A subset of B, then C subset of delta(M) (C) and delta(M) (A) subset of delta(M) (B). Let delta(M) be an expansion of submodules and delta(R) be an expansion of ideals. Then a proper submodule N is called a ( delta(R,M)-semiprimary) weakly delta(R,M) -semiprimary submodule of M if (rm is an element of N) 0(M) not equal rm is an element of N implies r is an element of delta(R)(N : M) or m is an element of delta(M) (N). Various results and examples concerning weakly delta(R,M)-semiprimary submodules are given.