On weakly $$\delta _{R,M}$$-semiprimary submodules

Abstract

Let R be a commutative ring with $$1\ne 0$$ 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 $$rm\in N$$ for $$r\in R$$ and $$m \in M$$ , then $$r\in \sqrt{N:M}$$ or $$m\in \sqrt{N}$$ , where $$rad(N)=\sqrt{N}$$ 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 $$0_{M}\ne rm\in N$$ for $$r\in R$$ and $$m \in M$$ , then $$r\in \sqrt{N:M}$$ or $$m\in \sqrt{N}$$ . 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 $$\delta _{M}:S(M)\rightarrow S(M)$$ 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 $$A\subseteq B$$ , then $$C\subseteq \delta _{M}(C)$$ and $$\delta _{M}(A)\subseteq \delta _{M}(B)$$ . 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 $$(rm\in N)$$ $$0_{M}\ne rm\in N$$ implies $$r\in \delta _{R}(N:M)$$ or $$m\in \delta _{M}(N)$$ . Various results and examples concerning weakly $$\delta _{R,M}$$ -semiprimary submodules are given.