Some Properties of the Nil-Graphs of Ideals of Commutative Rings

Let R be a commutative ring with identity and let Nil(R) be the ideal of all 2 nilpotent elements of R. Let I(R) = {I : I is a non-trivial ideal of R and there exists a 3 non-trivial ideal J such that IJ ⊆ Nil(R)}. The nil-graph of ideals of R is defined as the 4 simple undirected graph AG N (R) whose vertex set is I(R) and two distinct vertices I and 5 J are adjacent if and only if IJ ⊆ Nil(R). In this paper, we study the planarity and genus of 6 AG N (R). In particular, we have characterized all commutative Artin rings R for which the 7 genus of AG N (R) is either zero or one.

Bulletin of the Malaysian Mathematical Sciences Society, 2015

Let R be a commutative ring with identity and Nil(R) be the set of nilpotent elements of R. The nil-graph of ideals of R is defined as the graph AG N (R) whose vertex set is {I : (0) = I R and there exists a non-trivial ideal J such that IJ ⊆ Nil(R)} and two distinct vertices I and J are adjacent if and only if IJ ⊆ Nil(R). Here, some graph properties of AG N (R) are studied. For instance, some bounds for the diameter, girth and the radius of AG N (R) are given. In case that AG N (R) is a finite graph, it is proved that the center and median of AG N (R) coincide. Furthermore, we determine when the edge chromatic number of AG N (R) equals its maximum degree. Also, for every ring R, it is shown that both the clique number and vertex chromatic number of AG N (R) equal n + t, where n is the number of minimal prime ideals of R and t is the number of non-zero ideals of R which are contained in Nil(R).

2012

Let R be a commutative ring and A(R) be the set of ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A(R) * = A \ {(0)} and two distinct vertices I and J are adjacent if and only if IJ = (0). We investigate commutative rings R whose annihilating-ideal graphs have positive genus γ(AG(R)). It is shown that if R is an Artinian ring such that γ(AG(R)) < ∞, then either R has only finitely many ideals or (R, m) is a Gorenstein ring with maximal ideal m and v.dim R/m m/m 2 = 2. Also, for any two integers g ≥ 0 and q > 0, there are only finitely many isomorphism classes of Artinian rings R satisfying the conditions: (i) γ(AG(R)) = g and (ii) |R/m| ≤ q for every maximal ideal m of R. Also, it is shown that if R is a non-domain Noetherian local ring such that γ(AG(R)) < ∞, then either R is a Gorenstein ring or R is an Artinian ring with only finitely many ideals.

2011

In this paper we continue our study of annihilating-ideal graph of commutative rings, that was introduced in Part I (see [4]). Let R be a commutative ring with A(R) its set of ideals with nonzero annihilator and Z(R) its set of zero divisors. The annihilating-ideal graph of R is defined as the (undirected) graph AG(R) that its vertices are A(R) * = A(R)\ {(0)} in which for every distinct vertices I and J, I −−−J is an edge if and only if IJ = (0). First, we study the diameter of AG(R). A complete characterization for the possible diameter is given exclusively in terms of the ideals of R when either R is a Noetherian ring or Z(R) is not an ideal of R. Next, we study coloring of annihilating-ideal graphs. Among other results, we characterize when either χ(AG(R)) ≤ 2 or R is reduced and χ(AG(R)) ≤ ∞. Also it is shown that for each reduced ring R, χ(AG(R)) = cl(AG(R)). Moreover, if χ(AG(R)) is finite, then R has a finite number of minimal primes, and if n is this number, then χ(AG(R)) = cl(AG(R)) = n. Finally, we show that for a Noetherian ring R, cl(AG(R)) is finite if and only if for every ideal I of R with I 2 = (0), I has finite number of R-submodules.

Let $R$ be a commutative ring and ${\Bbb{A}}(R)$ be the set of ideals with non-zero annihilators. The annihilating-ideal graph of $R$ is defined as the graph ${\Bbb{AG}}(R)$ with vertex set ${\Bbb{A}}(R)^*={\Bbb{A}}\setminus\{(0)\}$ such that two distinct vertices $I$ and $J$ are adjacent if and only if $IJ=(0)$. We characterize commutative Noetherian rings $R$ whose annihilating-ideal graphs have finite genus $\gamma(\Bbb{AG}(R))$. It is shown that if $R$ is a Noetherian ring such that $0&lt;\gamma(\Bbb{AG}(R))&lt;\infty$, then $R$ has only finitely many ideals.

Algebra Colloquium, 2014

Let R be a commutative ring and Max (R) be the set of maximal ideals of R. The regular digraph of ideals of R, denoted by − − → Γreg(R), is a digraph whose vertex set is the set of all non-trivial ideals of R and for every two distinct vertices I and J, there is an arc from I to J whenever I contains a J-regular element. The undirected regular (simple) graph of ideals of R, denoted by Γreg(R), has an edge joining I and J whenever either I contains a J-regular element or J contains an I-regular element. Here, for every Artinian

2011

Let R be a commutative ring with A(R) its set of ideals with nonzero annihilator. In this paper and its sequel, we introduce and investigate the annihilatingideal graph of R, denoted by AG(R). It is the (undirected) graph with vertices A(R) * := A(R) \ {(0)}, and two distinct vertices I and J are adjacent if and only if IJ = (0). First, we study some finiteness conditions of AG(R). For instance, it is shown that if R is not a domain, then AG(R) has ACC (resp., DCC) on vertices if and only if R is Noetherian (resp., Artinian). Moreover, the set of vertices of AG(R) and the set of nonzero proper ideals of R have the same cardinality when R is either an Artinian or a decomposable ring. This yields for a ring R, AG(R) has n vertices (n ≥ 1) if and only if R has only n nonzero proper ideals. Next, we study the connectivity of AG(R). It is shown that AG(R) is a connected graph and diam(AG)(R) ≤ 3 and if AG(R) contains a cycle, then g(AG(R)) ≤ 4. Also, rings R for which the graph AG(R) is complete or star, are characterized, as well as rings R for which every vertex of AG(R) is a prime (or maximal) ideal. In Part II we shall study the diameter and coloring of annihilating-ideal graphs.

2020

The rings considered in this article are commutative with identity which admit at least one nonzero proper ideal. Let R be a ring. Recall that the intersection graph of ideals of R, denoted by G(R), is an undirected simple graph whose vertex set is the set of all nontrivial ideals of R (an ideal I of R is said to be nontrivial if I / ∈ {(0), R}) and distinct vertices I, J are joined by an edge in G(R) if and only if I ∩ J 6= (0). Let r ∈ N. The aim of this article is to characterize rings R such that G(R) is either bipartite or 3-partite. MSC: 13A15.

2021

Let R be a commutative ring with non-zero identity. The annihilator-inclusion ideal graph of R , denoted by ξR, is a graph whose vertex set is the of allnon-zero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacentif and only if either Ann(I) ⊆ J or Ann(J) ⊆ I. In this paper, we investigate the basicproperties of the graph ξR. In particular, we showthat ξR is a connected graph with diameter at most three, andhas girth 3 or ∞. Furthermore, we determine all isomorphic classes of non-local Artinian rings whose annihilator-inclusion ideal graphs have genus zero or one.

2018

Let \(R\) be a commutative ring and \(Z(R)^*\) be its set of non-zero zero-divisors. The annihilator graph of a commutative ring \(R\) is the simple undirected graph \(\operatorname{AG}(R)\) with vertices \(Z(R)^*\), and two distinct vertices \(x\) and \(y\) are adjacent if and only if \(\operatorname{ann}(xy)\neq \operatorname{ann}(x)\cup \operatorname{ann}(y)\). The notion of annihilator graph has been introduced and studied by A. Badawi [7]. In this paper, we determine isomorphism classes of finite commutative rings with identity whose \(\operatorname{AG}(R)\) has genus less or equal to one.