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Classifying annihilating-ideal graphs of commutative artinian rings

Amanda R. Curtis, Alexander J. Diesl, Jane C. Rieck
2018 Communications in Algebra  
In this article we investigate the annihilating-ideal graph of a commutative ring, introduced by Behboodi and Rakeei in [BR11a] .  ...  Consequently, we completely determine the graphs with 6 or fewer vertices which can be realized as the annihilating-ideal graph of a commutative ring.  ...  Acknowledgments This research was partially supported by a Wellesley College Brachman Hoffman Small Grant.  ... 
doi:10.1080/00927872.2018.1439040 fatcat:yexk5pyjlbbhbg5lj6lvj6ghsa

Essential ideals represented by mod-annihilators of modules [article]

Rameez Raja, Shariefuddin Pirzada
2022 arXiv   pre-print
Let R be a commutative ring with unity, M be a unitary R-module and G a finite abelian group (viewed as a ℤ-module). The main objective of this paper is to study properties of mod-annihilators of M.  ...  Finally, we discuss the correspondence of essential ideals of R and vertices of the annihilating graphs realized by M over R.  ...  In [17] , the authors introduced annihilating graphs realized by modules over commutative rings known as full-annihilating, semi-annihilating and star-annihilating graphs, .  ... 
arXiv:2203.02463v1 fatcat:3igncl5ql5cz7mk7q4g5onyhai

The Annihilating-Ideal Graph of a Ring [article]

F. Aliniaeifard, M. Behboodi, Y. Li
2014 arXiv   pre-print
We define an annihilating-ideal graph of a ring as a special type of zero-divisor graph of a semigroup. We introduce two ways to define the zero-divisor graphs of semigroups.  ...  Let S be a semigroup with 0 and R be a ring with 1. We extend the definition of the zero-divisor graphs of commutative semigroups to not necessarily commutative semigroups.  ...  Undirected Annihilating-Ideal Graphs for Matrix Rings Over Commutative Rings In this section we investigate the undirected annihilating-ideal graphs of matrix rings over commutative rings.  ... 
arXiv:1411.4159v1 fatcat:zau7plshojafpm64g55u2gwmom

Zero-Divisor Graphs of Matrices Over Commutative Rings

Ivana Božić, Zoran Petrović
2009 Communications in Algebra  
Then we use these results to discuss the relation between the diameter of the zero-divisor graph of a commutative ring R and that of the matrix ring M n R .  ...  We investigate the properties of (directed) zero-divisor graphs of matrix rings.  ...  This work was partially supported by the Ministry of Science and Environmental Protection of the Republic of Serbia Project #144020.  ... 
doi:10.1080/00927870802465951 fatcat:yakja2bqobasjdcd2qodpr7h74

THE ANNIHILATING-IDEAL GRAPH OF A RING

FARID ALINIAEIFARD, MAHMOOD BEHBOODI, YUANLIN LI
2015 Journal of the Korean Mathematical Society  
Let S be a semigroup with 0 and R be a ring with 1. We extend the definition of the zero-divisor graphs of commutative semigroups to not necessarily commutative semigroups.  ...  For a ring R define a directed graph APOG(R) to be equal to Γ(IPO(R)), where IPO(R) is a semigroup consisting of all products of two one-sided ideals of R, and define an undirected graph APOG(R) to be  ...  Undirected annihilating-ideal graphs for matrix rings over commutative rings In this section we investigate the undirected annihilating-ideal graphs of matrix rings over commutative rings.  ... 
doi:10.4134/jkms.2015.52.6.1323 fatcat:wrlyck56jfgkln5jeeiau7lv5a

On Containment and Radical Graphs of a Commutative Ring

Saba Al-Kaseasbeh
2016 Journal of Mathematics and Statistics  
In this study, we consider various graphs with the vertices being sets of ideals of a commutative ring with identity, R.  ...  We apply a graphical approach to the study of ideal relationships in commutative rings and from this we obtain some properties of their graphs.  ...  Jim Coykendall for suggesting the research problem and anonymous referee for their very constructive suggestions that contributed to the improvement of this version of the paper.  ... 
doi:10.3844/jmssp.2016.308.311 fatcat:ozqcn4b6crh2ri2rvrqrfb64dq

On the genus of nil-graph of ideals of commutative rings

T. Tamizh Chelvam, K. Selvakumar, P. Subbulakshmi
2017 Arab Journal of Mathematical Sciences  
Let R be a commutative ring with identity and let Nil(R) be the ideal of all nilpotent elements of R.  ...  In particular, we have characterized all commutative Artin rings R for which the genus of AG N (R) is either zero or one. article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd  ...  This work is supported by the UGC-BSR One-time grant (F.19-109/2013(BSR)) of University Grants Commission, Government of India for the first author.  ... 
doi:10.1016/j.ajmsc.2016.09.004 fatcat:poexfahiwzgb5ijkwtp4zxooly

On realizing zero-divisor graphs of po-semirings [article]

Houyi Yu, Tongsuo Wu
2011 arXiv   pre-print
As applications, we classify commutative rings R whose annihilating-ideal graph AG(R) are either bipartite graphs or complete graphs with horns.  ...  In this paper, we determine bipartite graphs and complete graphs with horns, which are realizable as zero-divisor graphs of po-semirings.  ...  So we have Some results on annihilating-ideal graphs of commutative rings The annihilating-ideal graph AG(R) of a commutative ring R was introduced in [9] and further studied in [10, 1, 2, 3] .  ... 
arXiv:1106.0350v1 fatcat:p7ahj52cj5gs3psj4i4xyjo63e

ON RINGS WHOSE ANNIHILATING-IDEAL GRAPHS ARE BLOW-UPS OF A CLASS OF BOOLEAN GRAPHS

Jin Guo, Tongsuo Wu, Houyi Yu
2017 Journal of the Korean Mathematical Society  
In this paper, we characterize the annihilating-ideal graphs AG(R) that are blow-ups of strong Boolean graphs, complemented graphs and preatomic graphs respectively.  ...  For a finite or an infinite set X, let 2 X be the power set of X. A class of simple graph, called strong Boolean graph, is defined on the vertex set 2 X \ {X, ∅}, with M adjacent to N if M ∩ N = ∅.  ...  The authors express their sincere thanks to the referee for the valuable comments and suggestions, and for providing some constructive examples such as Examples 3.6 and 4.8, which improve the paper a lot  ... 
doi:10.4134/jkms.j160283 fatcat:5lxubkcp4rbqphyxbeeccaez7u

The Annihilating-Ideal Graph of Commutative Rings I [article]

Mahmood Behboodi, Zahra Rakeei
2011 arXiv   pre-print
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).  ...  In Part II we shall study the diameter and coloring of annihilating-ideal graphs.  ...  , the notion of "comaximal graph of commutative rings" is investigated).  ... 
arXiv:0808.3187v2 fatcat:qld46jo23fcwxcxexoejnjwrji

Diameter, girth and cut vertices of the graph of equivalence classes of zero-divisors

Blake Allen, Erin Martin, Eric New, Dane Skabelund
2012 Involve. A Journal of Mathematics  
We explore the properties of E (R), the graph of equivalence classes of zerodivisors of a commutative Noetherian ring R.  ...  In Section 4, we look at properties of the cut-vertices of E (R). Throughout, R will denote a commutative Noetherian ring.  ...  Acknowledgements This paper was written during the Research Experience for Undergraduates conducted at Brigham Young University in the summer of 2010.  ... 
doi:10.2140/involve.2012.5.51 fatcat:azph4jdh2naffbmjspoyiexkfy

The Annihilating-Ideal Graph of Commutative Rings II [article]

Mahmood Behboodi, Zahra Rakeei
2011 arXiv   pre-print
Let R be a commutative ring with A(R) its set of ideals with nonzero annihilator and Z(R) its set of zero divisors.  ...  In this paper we continue our study of annihilating-ideal graph of commutative rings, that was introduced in Part I (see [5]).  ...  Thus throughout the paper, R denotes a commutative ring with identity, Z(R) denotes the the set of all zero divisors of R and I(R) denotes the set of all proper ideals of R.  ... 
arXiv:0808.3189v2 fatcat:62zpojkvyrb7zlafyms4c3wo4m

On zero divisors and prime elements of po-semirings [article]

Tongsuo Wu, Dancheng Lu, Yuanlin Li
2014 arXiv   pre-print
Applications to the ideal structure of commutative rings are considered.  ...  For a po-semiring A with Z(A)=A∖{0,1}, it is proved that A has finitely many maximal elements if ACC holds either for elements of A or for principal annihilating ideals of A.  ...  Thus all known results on graph properties for the semigroup graph Γ(S) hold for the annihilating ideal graph AG(R).  ... 
arXiv:1106.0348v2 fatcat:scals3y5srarxefa5yitlpwqe4

Page 6004 of Mathematical Reviews Vol. , Issue 89K [page]

1989 Mathematical Reviews  
{Reviewer’s remarks: In the proofs of the two results given above the author considers the right annihilator P of J, which is a prime ideal.  ...  (Note that Q is commutative precisely if it is a semilattice.) Secondly assume ® to be a field.  ... 

Zero-Divisor Graphs of Polynomials and Power Series Over Commutative Rings

Michael Axtell, James Coykendall, Joe Stickles
2005 Communications in Algebra  
The concept of the graph of the zero-divisors of a ring was first introduced by Beck in [3] when discussing the coloring of a commutative ring.  ...  We recall several results of zero divisor graphs of commutative rings.  ...  Suppose x is a nonzero element of P 1 ∩ P 2 . Note that the annihilator of x necessarily contains all of P 1 \ P 2 and P 2 \ P 1 .  ... 
doi:10.1081/agb-200063357 fatcat:z5ddf7shufefxkqizjpxc7jmwi
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