1. ON THE REFINEMENT OF THE UNIT AND UNITARY CAYLEY GRAPHS OF RINGS

M. Rezagholibeigi; A. R. Naghipour

Volume 7, Issue 1 , Summer and Autumn 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

2. THE ZERO-DIVISOR GRAPH OF A MODULE

A. Naghipour

Volume 4, Issue 2 , Winter and Spring 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

3. GENERALIZED PRINCIPAL IDEAL THEOREM FOR MODULES

A.R. Naghipour

Volume 3, Issue 1 , Summer and Autumn 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

4. SOME RESULTS ON STRONGLY PRIME SUBMODULES

A.R. Naghipour

Volume 1, Issue 2 , Winter and Spring 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