The ideal-based zero-divisor graph of commutative chained rings

Abstract

Abstract. Let I be a proper ideal of a commutative ring R with 1 = 0. The ideal-based zero-divisor graph of R with respect to I, denoted by ΓI (R), is the (simple) graph with vertices { x ∈ R \ I | xy ∈ I for some y ∈ R \ I }, and distinct vertices x and y are adjacent if and only if xy ∈ I. In this paper, we study ΓI (R) for commutative rings R such that R/I is a chained ring.

Total Graph of a Commutative Semiring With Respect to Identity-Summand Elements

“…This notion was later redefined by Anderson and Livingston in [6]. Since then, there has been a lot of interest in this subject and various papers were published establishing different properties of these graphs as well as relations between graphs of various extensions.…”

Total Identity-Summand Graph of a Commutative Semiring With Respect to a Co-Ideal

“…3. Let R be a semiring and I be a subtractive co-ideal of R. If T is an additively closed subset of R such that T ∩ I = ∅, then = {S : T ⊆ S, S is an additively closed subset of R and S ∩ I = ∅} has a maximal element.…”

“…In his work all elements of the ring were vertices of the graph. In [3], Anderson and Livingston introduced and studied the zero-divisor graph whose vertices are the non-zero zero-divisors of a ring. In [2], the authors defined the total graph of a ring R to be the (undirected) graph T (Γ(R)) with all elements of R as vertices, and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R), where Z(R) is the zero divisors of R. Also they studied the subgraph T 0 (Γ(R)) of T (Γ(R)) with vertices R \ {0}.…”

The Identity-Summand Graph of Commutative Semirings

“…One of the associated graphs to a ring R is the zero-divisor graph; it is a simple graph with vertex set Z(R) \ {0}, and two vertices x and y are adjacent if and only if xy = 0 which is due to Anderson and Livingston [8]. This graph was first introduced by Beck, in [11], where all the elements of R are considered as the vertices.…”