On annihilator graph of a finite commutative ring (original) (raw)
Related papers
On the Annihilator Graph of a Commutative Ring
Communications in Algebra, 2013
Let R be a commutative ring with nonzero identity, Z R be its set of zero-divisors, and if a ∈ Z R , then let ann R a = d ∈ R da = 0. The annihilator graph of R is the (undirected) graph AG R with vertices Z R * = Z R \ 0 , and two distinct vertices x and y are adjacent if and only if ann R xy = ann R x ∪ ann R y. It follows that each edge (path) of the zero-divisor graph R is an edge (path) of AG R. In this article, we study the graph AG R. For a commutative ring R, we show that AG R is connected with diameter at most two and with girth at most four provided that AG R has a cycle. Among other things, for a reduced commutative ring R, we show that the annihilator graph AG R is identical to the zero-divisor graph R if and only if R has exactly two minimal prime ideals.
The annihilating-ideal graph of commutative rings II
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.
Some Properties of Annihilator Graph of a Commutative Ring
Abstract: Let R be a commutative ring with unity. Let Z(R) be the set of all zero-divisors of R. For x Z(R), let ann (x) {yR | yx 0} R . We define the annihilator graph of R, denoted by ANNG(R), as the undirected graph whose set of vertices is Z(R)* = Z(R) {0}, and two distinct vertices x and y are adjacent if and only if ann (xy) ann (x) ann (y) R R R . In this paper, we study the ring-theoretic properties of R and the graph-theoretic properties of ANNG(R). For a commutative ring R, we show that ANNG(R) is connected, the diameter of ANNG(R) is at most two and the girth of ANNG(R) is at most four provided that ANNG(R) has a cycle. For a reduced commutative ring R, we study some characteristics of the annihilator graph ANNG(R) related to minimal prime ideals of R. Moreover, for a reduced commutative ring R, we establish some equivalent conditions which describe when ANNG(R) is a complete graph or a complete bipartite graph or a star graph. Keywords: Annihilator graph, diameter, girth, zero-divisor graph. 2010 Mathematics Subject Classification: Primary 13A15; Secondary 05C25, 05C38, 05C40.
On the annihilator-ideal graph of commutative rings
Ricerche di Matematica, 2016
Let R be a commutative ring with nonzero identity. We denote by AG(R) the annihilator graph of R, whose vertex set consists of the set of nonzero zero-divisors of R, and two distint vertices x and y are adjacent if and only if ann(x) ∪ ann(y) = ann(x y), where for t ∈ R, we set ann(t) := {r ∈ R | rt = 0}. In this paper, we define the annihiator-ideal graph of R, which is denoted by A I (R), as an undirected graph with vertex set A * (R), and two distinct vertices I and J are adjacent if and only if ann(I) ∪ ann(J) = ann(I J). We study some basic properties of A I (R) such as connectivity, diameter and girth. Also we investigate the situations under which the graphs AG(R) and A I (R) are coincide. Moreover, we examin the planarity of the graph A I (R).
The Annihilating-Ideal Graph of Commutative Rings I
Journal of Algebra and Its Applications, 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 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...
On the coloring of the annihilating-ideal graph of a commutative ring
Discrete Mathematics, 2012
Suppose that R is a commutative ring with identity. Let A(R) be the set of all ideals of R with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A(R) * = A(R) \ {(0)} and two distinct vertices I and J are adjacent if and only if IJ = (0). In Behboodi and Rakeei [8], it was conjectured that for a reduced ring R with more than two minimal prime ideals, girth(AG(R)) = 3. Here, we prove that for every (not necessarily reduced) ring R, ω(AG(R)) ≥ |Min(R)|, which shows that the conjecture is true. Also in this paper, we present some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring. Among other results, it is shown that if the chromatic number of the zero-divisor graph is finite, then the chromatic number of the annihilating-ideal graph is finite too. We investigate commutative rings whose annihilating-ideal graphs are bipartite. It is proved that AG(R) is bipartite if and only if AG(R) is triangle-free.
The Classification of the Annihilating-Ideal Graphs of Commutative Rings
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
The annihilator ideal graph of a commutative ring
2016
Let be a commutative ring with nonzero identity and be a proper ideal of . The annihilator graph of with respect to , which is denoted by , is the undirected graph with vertex-set for some and two distinct vertices and are adjacent if and only if , where . In this paper, we study some basic properties of , and we characterise when is planar, outerplanar or a ring graph. Also, we study the graph , where is the ring of integers modulo .
2017
Let R be a commutative ring with nonzero identity, Z(R) be its set of zero-divisors, and if a ∈ Z(R), then let ann R (a) = {d ∈ R | da = 0}. The annihilator graph of R is the (undirected) graph AG(R) with vertices Z(R) * = Z(R) \ {0}, and two distinct vertices x and y are adjacent if and only if ann R (xy) = ann R (x) ∪ ann R (y). It follows that each edge (path) of the zero-divisor graph Γ(R) is an edge (path) of AG(R). The extended zero-divisor graph of R is the undirected (simple) graph EG(R) with the vertex set Z(R) * , and two distinct vertices x and y are adjacent if and only if either Rx ∩ ann R (y) = {0} or Ry ∩ ann R (x) = {0}. Hence it follows that the zero-divisor graph Γ(R) is a subgraph of EG(R). In this paper, we collect some properties (many are recent) of the two graphs AG(R) and EG(R).