##### ZERO-DIVISOR GRAPH OF THE RINGS OF REAL MEASURABLE FUNCTIONS WITH THE MEASURES

H. Hejazipour; A. R. Naghipour

Volume 9, Issue 2 , January 2022, , Pages 175-192
##### Abstract
Let $M(X, \mathcal{A}, \mu)$ be the ring of real-valued measurable functionson a measurable space $(X, \mathcal{A})$ with measure $\mu$.In this paper, we study the zero-divisor graph of $M(X, \mathcal{A}, \mu)$,denoted by $\Gamma(M(X, \mathcal{A}, \mu))$.We give the relationships among graph properties ...  Read More

##### ON THE REFINEMENT OF THE UNIT AND UNITARY CAYLEY GRAPHS OF RINGS

M. Rezagholibeigi; A. R. Naghipour

Volume 7, Issue 1 , September 2019, , Pages 51-68
##### Abstract
Let $R$ be a ring (not necessarily commutative) with nonzero identity. We define $\Gamma(R)$ to be the graph with vertex set $R$ in which two distinct vertices $x$ and $y$ are adjacent if and only if there exist unit elements $u,v$ of $R$ such that $x+uyv$ is a unit of $R$. In this paper, basic properties ...  Read More

##### THE ZERO-DIVISOR GRAPH OF A MODULE

A. Naghipour

Volume 4, Issue 2 , January 2017, , Pages 155-171
##### Abstract
Let R be a commutative ring with identity and M an R-module. In this paper, we associate a graph to M, sayΓ(RM), such that when M=R, Γ(RM) coincide with the zero-divisor graph of R. Many well-known results by D.F. Anderson and P.S. Livingston have been generalized for ...  Read More

##### GENERALIZED PRINCIPAL IDEAL THEOREM FOR MODULES

A.R. Naghipour

Volume 3, Issue 1 , September 2015, , Pages 23-30
##### Abstract
The Generalized Principal Ideal Theorem is one of the cornerstones of dimension theory for Noetherian rings. For an R-module M, we identify certain submodules of M that play a role analogous to that of prime ideals in the ring R. Using this definition, we extend the Generalized Principal Ideal Theorem ...  Read More

##### SOME RESULTS ON STRONGLY PRIME SUBMODULES

A.R. Naghipour

Volume 1, Issue 2 , January 2014, , Pages 79-89
##### Abstract
Let $R$ be a commutative ring with identity and let $M$ be an $R$-module. A proper submodule $P$ of $M$ is called strongly prime submodule if $(P + Rx : M)y P$ for $x, y M$, implies that $x P$ or $y P$. In this paper, we study more properties of strongly prime submodules. It is shown that a ...  Read More