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# 20 - M. Behboodi (with Z. Rakee), The annhilating-ideal graph of commutative rings I. J. Algebra Appl. 10 (4) (2011), 727-739. (ISI)

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 ACC (resp., DCC) on vertices if and only if R is Noetherian (resp., 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 gr(AG(R)) ≤ 4. Also, rings R for which the graph AG(R) is complete or star, are characterized, as well as rings R for which every vertex of AG(R) is a prime (or maximal) ideal. In Part II we shall study the diameter and coloring of annihilating-ideal graphs.

Key Words: Commutative rings; Annihilating-ideal; Zero-divisor; Graph

2000 Mathematics Subject Classiﬁcation: 13A15; 05C75.