On the Torsion Graph and Von Neumann Regular Rings

P. Malakooti Rad, Sh. Ghalandarzadeh, S. Shirinkam

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. A torsion graph of $M$, denoted by $\Gamma(M)$, is a graph whose vertices are the non-zero torsion elements of $M$, and two distinct vertices $x$ and $y$ are adjacent if and only if $[x:M][y:M]M=0$. We investigate the relationship between the diameters of $\Gamma(M)$ and $\Gamma(R)$, and give some properties of minimal prime submodules of a multiplication $R$-module $M$ over a von Neumann regular ring. In particular, we show that for a multiplication $R$-module $M$ over a B\'ezout ring $R$ the diameter of $\Gamma(M)$ and $\Gamma(R)$ is equal, where $M\neq T(M)$. Also, we prove that, for a faithful multiplication $R$-module $M$ with $|M| \neq 4$,$\Gamma(M)$ is a complete graph if and only if $\Gamma(R)$ is a complete graph. Siamak Yassemi 8486