The exact annihilating-ideal graph of a commutative ring

Yıl 2021, , 119 - 138, 20.05.2021

Subramanian Visweswaran Premkumar T. Lalchandani

https://doi.org/10.13069/jacodesmath.938105

Öz

The rings considered in this article are commutative with identity. For an ideal $I$ of a ring $R$, we denote the annihilator of $I$ in $R$ by $Ann(I)$. An ideal $I$ of a ring $R$ is said to be an exact annihilating ideal if there exists a non-zero ideal $J$ of $R$ such that $Ann(I) = J$ and $Ann(J) = I$. For a ring $R$, we denote the set of all exact annihilating ideals of $R$ by $\mathbb{EA}(R)$ and $\mathbb{EA}(R)\backslash \{(0)\}$ by $\mathbb{EA}(R)^{*}$. Let $R$ be a ring such that $\mathbb{EA}(R)^{*}\neq \emptyset$. With $R$, in [Exact Annihilating-ideal graph of commutative rings, {\it J. Algebra and Related Topics} {\bf 5}(1) (2017) 27-33] P.T. Lalchandani introduced and investigated an undirected graph called the exact annihilating-ideal graph of $R$, denoted by $\mathbb{EAG}(R)$ whose vertex set is $\mathbb{EA}(R)^{*}$ and distinct vertices $I$ and $J$ are adjacent if and only if $Ann(I) = J$ and $Ann(J) = I$. In this article, we continue the study of the exact annihilating-ideal graph of a ring. In Section 2 , we prove some basic properties of exact annihilating ideals of a commutative ring and we provide several examples. In Section 3, we determine the structure of $\mathbb{EAG}(R)$, where either $R$ is a special principal ideal ring or $R$ is a reduced ring which admits only a finite number of minimal prime ideals.

Anahtar Kelimeler

Exact annihilating ideal, Exact annihilating-ideal graph, Connectedness, Reduced ring, Special principal ideal ring

Kaynakça

• [1] G. Aalipour, S. Akbari, M. Behboodi, R. Nikandish, M. J. Nikmehr, F. Shaveisi, The classification of annihilating-ideal graphs of commutative rings, Algebra Colloq. 21(2) (2014) 249–256.
• [2] G. Aalipour, S. Akbari, R. Nikandish, M. J. Nikmehr, F. Shaveisi, On the coloring of the annihilating ideal graph of a commutative ring, Discrete Math. 312 (2012) 2620–2626.
• [3] D. D. Anderson, M. Naseer, Beck’s coloring of a commutative ring, J. Algebra 159 (1993) 500–514.
• [4] M. F. Atiyah and I.G. Macdonald, Introduction to Commutative Algebra, Addison Wesley, Reading, Massachusetts (1969).
• [5] R. Balakrishnan and K. Ranganathan, A Textbook of Graph Theory, Universitext, Springer, New York (2000).
• [6] I. Beck, Coloring of commutative rings, J. Algebra 116(1) (1988) 208–226.
• [7] M. Behboodi, Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl. 10(4) (2011) 727–739.
• [8] M. Behboodi, Z. Rakeei, The annihilating-ideal graph of commutative rings II, J. Algebra Appl. 10(4) (2011) 741–753.
• [9] N. Deo, Graph Theory with Applications to Engineering and Computer Science, Prentice-Hall of India Private Limited, New Delhi (1994).
• [10] R. Gilmer, Multiplicative Ideal Theory, Marcel-Dekker, New York (1972).
• [11] M. Hadian, Unit action and the geometric zero-ideal ideal graph, Comm. Algebra 40(8) (2012) 2920–2931.
• [12] I. B. Henriques and L. N. Sega, Free resolution over short Gorsentein local rings, Math. Z. 267 (2011) 645–663.
• [13] N. Jacobson, Basic Algebra II, Hindustan Publishing Corporation, Delhi (1984).
• [14] I. Kaplansky, Commutative Rings, The University of Chicago Press, Chicago (1974).
• [15] P. T. Lalchandani, Exact zero-divisor graph, Int. J. Sci. Engg. and Mang. 1(6) (2016) 14–17.
• [16] P. T. Lalchandani, Exact zero-divisor graph of a commutative ring, Int. J. Math. Appl. 6(4) (2018) 91–98.
• [17] P. T. Lalchandani, Exact annihilating-ideal graph of commutative rings, J. Algebra and Related Topics 5(1) (2017) 27–33.
• [18] D. G. Northcott, Lessons on rings, modules and multiplicities, Cambridge University Press, Cambridge (1968).
• [19] S. Visweswaran and P. Sarman, On the complement of a graph associated with the set of all nonzero annihilating ideals of a commutative ring, Discrete Math. Algorithms Appl. 8(3) (2016) Article ID:1650043 22 pages.