The first part of the paper is concerned to relationship between the sets of associated primes of the generalized $d$-local cohomology modules and the ordinary  generalized local cohomology  modules.  Assume that $R$ is a commutative Noetherian local ring, $M$ and $N$  are  finitely generated  $R$-modules and $d, t$ are two integers. We prove that $Ass H^t_d(M,N)=bigcup_{Iin Phi} Ass H^t_I(M,N)...

In this paper, by using elementary tools of commutative algebra,we prove the persistence property for two especial classes of rings. In fact, thispaper has two main sections. In the first main section, we let R be a Dedekindring and I be a proper ideal of R. We prove that if I1, . . . , In are non-zeroproper ideals of R, then Ass1(Ik11 . . . Iknn ) = Ass1(Ik11 ) [ · · · [ Ass1(Iknn )for all k1,...

The total graph of a commutative ring have been introduced and studied by D. F. Anderson and A. Badawi in [1]. In a manner analogous to a commutative ring, the total torsion element graph of a module M over a commtative ring R can be defined as the undirected graph T (Γ(M)). The basic properties and possible structures of the graph T (Γ(M)) are studied. The main purpose of this paper is to exte...

The concept of a zero-divisor graph of a commutative ring was first introduced in Beck (1988), and later redefined in Anderson and Livingston (1999). Redmond (2002) further extended this concept to the noncommutative case, introducing several definitions of a zero-divisor graph of a noncommutative ring. Recently, the diameter and girth of polynomial and power series rings over a commutative rin...

In this paper, two outwardly different graphs, namely, the zero-divisor graph [Formula: see text] and comaximal of ring all real-valued continuous functions having countable range, defined on any zero-dimensional space text], are investigated. It is observed that these graphs exhibit resemblance, so far as diameters, girths, connectedness, triangulatedness or hypertriangulatedness concerned. Ho...

This paper describes an application of research that sits at the intersection of commutative algebra and combinatorics: Betti numbers of graphs. In particular, we describe a correspondence between simple undirected graphs and a class of ideals in a polynomial ring. We then briefly introduce some of the algebraic invariants that can be associated to the ideal and the relation of these invariants...

Let R be a commutative ring and let ? Z n the zero divisor graph of id="M2"> R , whose vertices are nonzero divisors id="M3"> such that two id="M4"> u , v adjacen...

an ideal i of a ring r is called right baer-ideal if there exists an idempotent e 2 r such that r(i) = er. we know that r is quasi-baer if every ideal of r is a right baer-ideal, r is n-generalized right quasi-baer if for each i e r the ideal in is right baer-ideal, and r is right principaly quasi-baer if every principal right ideal of r is a right baer-ideal. therefore the concept of baer idea...