Let R be a commutative ring with Z(R) , its set of zero-divisors and Reg(R) , its set of regular elements. Total graph of R , denoted by T (Γ(R)) , is the graph with all elements of R as vertices, and two distinct vertices x, y ∈ R , are adjacent in T (Γ(R)) if and only if x+ y ∈ Z(R) . In this paper, some properties of T (Γ(R)) have been investigated, where R is a finite commutative ring and a...

Let M be an Artinian module over the commutative ring A (with nonzero identity) and a p spec A be such that a is a finitely generated ideal of A and aM = M. Also suppose that H = H where H. = M/ (0: a )for i

let $r$ be a commutative ring with identity and $m$ an $r$-module. in this paper, we associate a graph to $m$, say ${gamma}({}_{r}m)$, such that when $m=r$, ${gamma}({}_{r}m)$ coincide with the zero-divisor graph of $r$. many well-known results by d.f. anderson and p.s. livingston have been generalized for ${gamma}({}_{r}m)$. we show that ${gamma}({}_{r}m)$ is connected with ${diam}({gamma}({}_...

Let R be a commutative integral domain with quotient field K and let P be a nonzero strongly prime ideal of R. We give several characterizations of such ideals. It is shown that (P : P) is a valuation domain with the unique maximal ideal P. We also study when P^{&minus1} is a ring. In fact, it is proved that P^{&minus1} = (P : P) if and only if P is not invertible. Furthermore, if P is invertib...

Let R be a commutative ring with identity and let I be an ideal of R. Let R 1 I be the subring of R×R consisting of the elements (r,r + i) for r ∈ R and i ∈ I. We study the diameter and girth of the zero-divisor graph of the ring R 1 I.

Throughout this paper, R will denote a commutative ring with identity and M is a unitary R- module and Z will denote the ring of integers. We introduce the graph Ω(M) of module M with the set of vertices contain all nontrivial non-essential submodules of M. We investigate the interplay between graph-theoretic properties of Ω(M) and algebraic properties of M. Also, we assign the values of natura...

Let $(R,m)$ be a commutative Noetherian local ring, $M$ a finitely generated $R$-module of dimension $d$, and let $I$ be an ideal of definition for $M$. In this paper, we extend cite[Corollary 10(4)]{P} and also we show that if $M$ is a Cohen-Macaulay $R$-module and $d=2$, then $lambda(frac{widetilde{I^nM}}{Jwidetilde{I^{n-1}M}})$ does not depend on $J$ for all $ngeq 1$, where $J$ is a minimal ...

Let R be a commutative Noetherian ring, I an ideal of R and M a non-zero R-module. In this paper we calculate the extension of annihilator of local cohomology modules H^t_I(M), t≥0, under the ring extension R⊂R[X] (resp. R⊂R[[X]]). By using this extension we will present some of the faithfulness conditions of local cohomology modules, and show that if the Lynch's conjecture, i...