# The exact annihilating-ideal graph of a commutative ring

### Abstract

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.

### References

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.

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.

D. D. Anderson, M. Naseer, Beck’s coloring of a commutative ring, J. Algebra 159 (1993) 500–514.

M. F. Atiyah and I.G. Macdonald, Introduction to Commutative Algebra, Addison Wesley, Reading, Massachusetts (1969).

R. Balakrishnan and K. Ranganathan, A Textbook of Graph Theory, Universitext, Springer, New York (2000).

I. Beck, Coloring of commutative rings, J. Algebra 116(1) (1988) 208–226.

M. Behboodi, Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl. 10(4) (2011) 727–739.

M. Behboodi, Z. Rakeei, The annihilating-ideal graph of commutative rings II, J. Algebra Appl. 10(4) (2011) 741–753.

N. Deo, Graph Theory with Applications to Engineering and Computer Science, Prentice-Hall of India Private Limited, New Delhi (1994).

R. Gilmer, Multiplicative Ideal Theory, Marcel-Dekker, New York (1972).

M. Hadian, Unit action and the geometric zero-ideal ideal graph, Comm. Algebra 40(8) (2012) 2920–2931.

I. B. Henriques and L. N. Sega, Free resolution over short Gorsentein local rings, Math. Z. 267 (2011) 645–663.

N. Jacobson, Basic Algebra II, Hindustan Publishing Corporation, Delhi (1984).

I. Kaplansky, Commutative Rings, The University of Chicago Press, Chicago (1974).

P. T. Lalchandani, Exact zero-divisor graph, Int. J. Sci. Engg. and Mang. 1(6) (2016) 14–17.

P. T. Lalchandani, Exact zero-divisor graph of a commutative ring, Int. J. Math. Appl. 6(4) (2018) 91–98.

P. T. Lalchandani, Exact annihilating-ideal graph of commutative rings, J. Algebra and Related Topics 5(1) (2017) 27–33.

D. G. Northcott, Lessons on rings, modules and multiplicities, Cambridge University Press, Cambridge (1968).

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.