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Want more? Advanced embedding details, examples, and help! In this paper we continue our study of annihilating-ideal graph of commutative rings, that was introduced in Part I see .
Next, we study coloring of annihilating-ideal graphs. Addeddate External-identifier urn:arXiv There are no reviews yet. Be the first one to write a review. Additional Collections.Let R be a commutative ring. In this paper, we introduce and study the compressed annihilator graph of R.
For a reduced ring Rwe show that compressed annihilator graph of R is identical to the compressed zero-divisor graph of R if and only if 0 is a 2-absorbing ideal of R. As a consequence, we show that an Artinian ring R is either local or reduced whenever 0 is a 2-absorbing ideal of R.
This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Anderson and J. LaGrange, Commutative Boolean monoids, reduced rings, and the compressed zero-divisor graph, J. Pure Appl. Algebra, — Anderson and P. Livingston, The zero-divisor graph of a commutative ring, J. Atiyah and I. Macdonald, Introduction to commutative algebra, Addison-Wesly Google Scholar.
Badawi, On 2-absorbing ideals of commutative rings, Bull. Badawi, On the annihilator graph of a commutative ring, Comm. Algebra42— Beck, Coloring of commutative rings, J.
Sather-Wagstaff, L. Sheppardson and S. Francisco et al. Mulay, Cycles and symmetries of zero-divisors, Comm. Algebra30—Read this paper on arXiv. 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 Rdenoted by A G R. First, we study some finiteness conditions of A G R. Moreover, the set of vertices of A G R and the set of nonzero proper ideals of R have the same cardinality when R is either an Artinian or a decomposable ring. Next, we study the connectivity of A G R.
Also, rings R for which the graph A G R is complete or star, are characterized, as well as rings R for which every vertex of A G R is a prime or maximal ideal. In Part II we shall study the diameter and coloring of annihilating-ideal graphs. In the literature, there are many papers on assigning a graph to a ring, a group, semigroup or a module see for example [,].
In fact, the concept of the zero divisor graph of a commutative ring R was first introduced by Beck , where he was mainly interested in colorings. In his work all elements of the ring were vertices of the graph. This investigation of colorings of a commutative ring was then continued by Anderson and Naseer in . Let Z R be the set of zero-divisors of R.
Throughout this paper R will be a commutative ring with identity. We denote the set of all proper ideals of R by I R. We investigate the interplay between the graph-theoretic properties of A G R and the ring-theoretic properties of R. In Section 1, we study some finiteness conditions of annihilating-ideal graphs.
These facts motivates us to the following conjecture: for a non-domain ring Rthe set of vertices of A G R and the set of nonzero proper ideals of R have the same cardinality. The conjecture above is true for all Artinian rings as well as all decomposable rings see Proposition 1. Recall that a graph G is connected if there is a path between every two distinct vertices. A graph in which each pair of distinct vertices is joined by an edge is called a complete graph.
Also, if a graph G contains one vertex to which all other vertices are joined and G has no other edges, is called a star graph.
In Section 2, the connectivity of the annihilating-ideal graphs are studied. Also, rings R for which the graph A G R is a complete or star graph are characterized, as well as rings R for which every vertex of A G R is a prime or maximal ideal. In part II we shall continue the study of this construction via diameter and coloring.
Finiteness conditions of annihilating-ideal graphs.In mathematicsspecifically module theorythe annihilator of a moduleor a subset of a module, is a concept generalizing torsion and orthogonality. Let R be a ringand let M be a left R - module. Choose a non-empty subset S of M.
It is the set of all elements of R that "annihilate" S the elements for which S is a torsion set.
On the planarity and perfectness of annihilator ideal graphs
If the ring R can be understood from the context, the subscript R can be omitted. Since R is a module over itself, S may be taken to be a subset of R itself, and since R is both a right and a left R module, the notation must be modified slightly to indicate the left or right side.
If R is commutativethen the equality holds. Recall that the support of a module is defined as. Using the relation to support, this gives the relation with the annihilator . Then, the annihilator of a finite module is non-trivial only if it is entirely torsion.
This is because. In fact the annihilator of a torsion module.Every maximal Ideal of a commutative ring with unity is a prime ideal.
This shows the annihilators can be easily be classified over the integers. It is interesting to study rings for which this lattice or its right counterpart satisfy the ascending chain condition or descending chain condition. If either lattice has either of these chain conditions, then R has no infinite orthogonal sets of idempotents. Lam In particular:. Given a module M over a Noetherian commutative ring Ra prime ideal of R that is an annihilator of a nonzero element of M is called an associated prime of M.
From Wikipedia, the free encyclopedia.
The Annihilating-Ideal Graph of Commutative Rings I
Concept in module theory. This article has multiple issues. Please help improve it or discuss these issues on the talk page. Learn how and when to remove these template messages. This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources.
Unsourced material may be challenged and removed. This article may be confusing or unclear to readers. Please help us clarify the article. There might be a discussion about this on the talk page. June Learn how and when to remove this template message. Retrieved Categories : Ideals Module theory Ring theory. Hidden categories: Articles with short description Articles needing additional references from January All articles needing additional references Wikipedia articles needing clarification from June All Wikipedia articles needing clarification Articles with multiple maintenance issues Harv and Sfn no-target errors.An elementary annihilator of a ring is an annihilator that has the form.
We define the elementary annihilator dimension of the ringdenoted byto be the upper bound of the set of all integers such that there is a chain of annihilators of.
We use this dimension to characterize some zero-divisors graphs. Let be a ring and be a nonempty subset of. We call the annihilator of in denoted by or the set. If is a singleton then will be denoted by. If then is called an elementary annihilator. An annihilator is said to be maximal if it is maximal in the set of all proper annihilators of. It is well known that all maximal annihilators are elementary. For an elementary annihilator chain is said to be a chain of elementary annihilators with length ending in.
The upper bound of the set of all lengths of elementary annihilator chains ending in is called the elementary annihilator height of or. In this paper, we introduce a dimension of a ring using elementary annihilator chains called elementary annihilator dimensiondenoted by.
The is the upper bound of the set of elementary annihilator heights. We use this dimension to study zero-divisor graphs. We introduce a class of rings called isometric maximal elementary annihilator rings, in short IMEA -rings. That is the class of rings with finite EAdimension whose all maximal annihilators have the same height. Definition 1. One says that this chain is an elementary annihilator chain of length ending in.
One defines the elementary annihilator height ofdenoted byas the upper bound of the set of all lengths of elementary annihilator chains ending in.
Example 2. Indeed, is the longest chain of elementary annihilators in. Indeed, is a chain of length one. Remark 3. We denote by the set of all nilpotent elements of. Theorem 4. Let and be its index of nilpotency; one has:. If or is infinite the result is obvious. Otherwise, there exists a chain whose length is and it ends in.
Let be this chain.
The Annihilating-Ideal Graph of Commutative Rings II
Moreover, we have. Corollary 5. If satisfies is finite then. In particular, if then for all. Theorem 6. Let and be two rings; then; 1 is finite if and only if and are finite.
Let be a nonzero zero-divisor. If and are nonzero then. If one of them is zero, for example, then.Let R be a commutative ring with nonzero identity. This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Aalipour, G. Discrete Math. Aliniaeifard, F. Algebra 41— Atiyah, M. Addision-Wesley Publishing Company, London Google Scholar.
Anderson, D. Algebra— Badawi, A. Algebra 421—14 Beck, I. Behboodi, M. Algebra Appl. Chakrabarty, I. DeMeyer, F. Semigroup Forum 65— Gilter, I. McLean, K.
Redmond, S. Download references. The authors are deeply grateful to the referee for careful reading of the manuscript and helpful suggestions. BoxMashhad, Iran. Department of Mathematics, University of Neyshabur, P.To browse Academia. Skip to main content. Log In Sign Up. IOSR Journals. Volume 10, Issue 5 Ver. IV Sep-Oct. Let Z R be the set of all zero-divisors 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. Introduction Let R be a commutative ring with unity, and Z R be its set of all zero-divisors. The concept of a zero-divisor graph of a commutative ring R was first introduced by I. Beck in , where all the elements of the ring R were taken as the vertices of the graph.
In , D. Anderson and P. In , A. In this paper, we give the definition of the annihilator graph in another way. Badawi in  if and only if R has exactly two minimal prime ideals. For the sake of completeness, we state some definitions and notations used throughout this paper.
Let G be an undirected graph. We say that G is connected if there exists a path between any two distinct vertices. A subgraph of G is a graph having all of its points and lines in G. A spanning subgraph is a subgraph containing all the vertices of G. The girth of G, denoted by gr Gis the length of a shortest cycle in G if G contains no cycle, www.
We denote by C the graph consisting of a cycle with n vertices. A graph G is complete if any n two distinct vertices are adjacent. The complete graph with n vertices will be denoted by K we allow n to be an infinite cardinal.
A complete bipartite graph is a graph G which may be partitioned into two disjoint nonempty vertex sets A and B such that two distinct vertices are adjacent if and only if they are in distinct vertex sets.
If one of the vertex set is singleton, we call G is a star graph. As usual, the ring of integers and the ring of integers modulo n will be denoted by Z and Z nrespectively.