A. R. Naghipour
Department of Mathematical Sciences, Shahrekord University, P.O. Box 115, Shahrekord, Iran.
[ 1 ] - When is the ring of real measurable functions a hereditary ring?
Let $M(X, mathcal{A}, mu)$ be the ring of real-valued measurable functions on a measure space $(X, mathcal{A}, mu)$. In this paper, we characterize the maximal ideals in the rings of real measurable functions and as a consequence, we determine when $M(X, mathcal{A}, mu)$ is a hereditary ring.
[ 2 ] - THE ZERO-DIVISOR GRAPH OF A MODULE
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 Γ(RM). We Will show that Γ(RM) is connected withdiam Γ(RM)≤ 3 and if Γ(RM) contains a cycle, then Γ(RM)≤4. We will also show tha...
[ 3 ] - GENERALIZED PRINCIPAL IDEAL THEOREM FOR MODULES
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 to modules.
[ 4 ] - SOME RESULTS ON STRONGLY PRIME SUBMODULES
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 $(P + Rx : M)ysubseteq P$ for $x, yin M$, implies that $xin P$ or $yin P$. In this paper, we study more properties of strongly prime submodules. It is shown that a finitely generated $R$-module $M$ is Artinian if and only if $M$ is Noetherian and every st...
[ 5 ] - ON THE REFINEMENT OF THE UNIT AND UNITARY CAYLEY GRAPHS OF RINGS
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 of $Gamma(R)$ are studied. We investigate connectivity and the girth of $Gamma(R)$, where $...
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