Narrow Results

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 annihilating-ideal 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 ascending chain condition (respectively, descending chain condition) on vertices if and only if R is Noetherian (respectively, 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 gr(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.

In this paper we continue our study of annihilating-ideal graph of commutative rings, that was introduced in (The annihilating-ideal graph of commutative rings I, to appear in J. Algebra Appl.). Let R be a commutative ring with 𝔸(R) be 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 I² = (0), I has finite number of R-submodules.

Let $R$ be a commutative ring with identity. We consider a simple graph associated with $R$, denoted by ${\mathrm{\Omega }}_R^{\ast }$, whose vertex set is the set of all non-trivial ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent whenever $J\text{Ann}(I)=(0)$ or $I\text{Ann}(J)=(0)$. In this paper, we characterize the commutative Artinian non-local ring $R$ for which ${\mathrm{\Omega }}_R^{\ast }$ has genus one and crosscap one.

Let $R$ be a commutative ring with identity and Nil$(R)$ be the ideal consisting of all nilpotent elements of $R$. Let $I(R)=\{I:I\text{ is a non-trivial ideal of }R\text{ and there exists}$$\text{ a non-trivial ideal}$$J\text{ such that }IJ\subseteq \mathrm{\text{Nil}}(R)\}$. The nil-graph of ideals of $R$ is defined as the graph ${𝔸𝔾}_N(R)$ whose vertex set is the set $I(R)$ and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ\subseteq \mathrm{\text{Nil}}(R)$. In this paper, we discuss some properties of nil-graph of ideals concerning connectedness, split and claw free. Also we characterize all commutative Artinian rings $R$ for which the nil-graph ${𝔸𝔾}_N(R)$ has genus 2.