Prime submodule sum graph of a module over a commutative ring

Abstract

Let $R$ be a commutative ring with unity and $M$ be an $R$-module. The prime submodule sum graph of $M$, denoted by $PSS(M)$, is an undirected graph whose vertices are nonzero proper submodules of $M$ and two distinct vertices $N$ and $P$ are adjacent if and only if $N + P$ is a prime submodule of $M$. In this paper, we study some basic graph theoretic properties of the graph $PSS(M)$. We also discuss some results on homomorphisms of $PSS(M)$. Finally, planarity and domination numbers of $PSS(M)$ are also investigated.