In this article, we give several generalizations of the concept of annihilating an ideal graph over a commutative ring with identity to modules. We observe that, over a commutative ring, R, AG∗(RM) is connected, and diamAG∗(RM) ≤ 3. Moreover, if AG∗(RM) contains a cycle, then grAG∗(RM) ≤ 4. Also for an R-module M with A∗(M) ̸= S(M) \ {0}, A∗(M) = ∅, if and only if M is a uniform module, and ann(...

Ž . Ž . Let R be a commutative ring with 1 and let Z R be its set of Ž . Ž . zero-divisors. We associate a simple graph G R to R with vertices Ž . Ž . 4 Z R * s Z R y 0 , the set of nonzero zero-divisors of R, and for disŽ . tinct x, y g Z R *, the vertices x and y are adjacent if and only if xy s 0. Ž . Thus G R is the empty graph if and only if R is an integral domain. The main object of this...

For any reduced commutative $f$-ring with identity and bounded inversion, we show that a condition which is obviously necessary for the socle of the ring to coincide with the socle of its bounded part, is actually also sufficient. The condition is that every minimal ideal of the ring consist entirely of bounded elements. It is not too stringent, and is satisfied, for instance, by rings of ...

Abstract Let R be a commutative ring with identity which is not an integral domain. An ideal I of a ring R is called an annihilating ideal if there exists r ∈ R r {0} such that Ir = (0). In this paper, we consider a simple undirected graph associated with R denoted by Ω(R) whose vertex set equals the set of all nonzero annihilating ideals of R and two distinct vertices I, J are adjacent if and ...

Let R be a commutative ring with identity. An ideal I of a ring R is called an annihilating ideal if there exists r ∈ R \ {0} such that Ir = (0) and an ideal I of R is called an essential ideal if I has non-zero intersection with every other non-zero ideal of R. The sum-annihilating essential ideal graph of R, denoted by AER, is a graph whose vertex set is the set of all non-zero annihilating i...

Let R be a commutative noetherian ring and Γ a finite group. In this paper,we study Gorenstein homological dimensions of modules with respect to a semi-dualizing module over the group ring  . It is shown that Gorenstein homological dimensions  of an  -RΓ module M with respect to a semi-dualizing module,  are equal over R and RΓ  .

A ring $R$ is a strongly clean ring if every element in $R$ is the sum of an idempotent and a unit that commutate. We construct some classes of strongly clean rings which have stable range one. It is shown that such cleanness of $2 imes 2$ matrices over commutative local rings is completely determined in terms of solvability of quadratic equations.

let r be a commutative noetherian ring. we study the behavior of injectiveand at dimension of r-modules under the functors homr(-,-) and -×r-.