A GENERALIZATION OF THE ESSENTIAL GRAPH FOR MODULES OVER COMMUTATIVE RINGS

Authors : F SOHEILNIA, Sh PAYROVI, A BEHTOEI

Pages : 211-222

Doi:10.24330/ieja.852234

View : 15 | Download : 12

Publication Date : 2021-01-05

Article Type : Research Paper

Abstract :Let $R$ be a commutative ring with nonzero identity and let $M$ be a unitary $R$-module. The essential graph of $M$, denoted by $EGinsert ignore into journalissuearticles values(M);$ is a simple undirected graph whose vertex set is $Zinsert ignore into journalissuearticles values(M);\setminus {\rm Ann}_Rinsert ignore into journalissuearticles values(M);$ and two distinct vertices $x$ and $y$ are adjacent if and only if ${\rm Ann}_{M}insert ignore into journalissuearticles values(xy);$ is an essential submodule of $M$. Let $rinsert ignore into journalissuearticles values({\rm Ann}_Rinsert ignore into journalissuearticles values(M););\not={\rm Ann}_Rinsert ignore into journalissuearticles values(M);$. It is shown that $EGinsert ignore into journalissuearticles values(M);$ is a connected graph with ${\rm diam}insert ignore into journalissuearticles values(EGinsert ignore into journalissuearticles values(M););\leq 2$. Whenever $M$ is Noetherian, it is shown that $EGinsert ignore into journalissuearticles values(M);$ is a complete graph if and only if either $Zinsert ignore into journalissuearticles values(M);=rinsert ignore into journalissuearticles values({\rm Ann}_Rinsert ignore into journalissuearticles values(M););$ or $EGinsert ignore into journalissuearticles values(M);=K_{2}$ and ${\rm diam}insert ignore into journalissuearticles values(EGinsert ignore into journalissuearticles values(M););= 2$ if and only if there are $x, y\in Zinsert ignore into journalissuearticles values(M);\setminus {\rm Ann}_Rinsert ignore into journalissuearticles values(M);$ and $\frak p\in{\rm Ass}_Rinsert ignore into journalissuearticles values(M);$ such that $xy\not \in \frak p$. Moreover, it is proved that ${\rm gr}insert ignore into journalissuearticles values(EGinsert ignore into journalissuearticles values(M););\in \{3, \infty\}$. Furthermore, for a Noetherian module $M$ with $rinsert ignore into journalissuearticles values({\rm Ann}_Rinsert ignore into journalissuearticles values(M););={\rm Ann}_Rinsert ignore into journalissuearticles values(M);$ it is proved that $|{\rm Ass}_Rinsert ignore into journalissuearticles values(M);|=2$ if and only if $EGinsert ignore into journalissuearticles values(M);$ is a complete bipartite graph that is not a star.
Keywords : Prime submodule, essential submodule, essential graph