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The annihilating-ideal graph of commutative rings i

机译：交换环i的灭理想图

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Let R be a commutative ring, with (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 (R). It is the (undirected) graph with vertices (R)*:= (R){(0)}, and two distinct vertices I and J are adjacent if and only if IJ = (0). First, we study some finiteness conditions of (R). For instance, it is shown that if R is not a domain, then (R) has ascending chain condition (respectively, descending chain condition) on vertices if and only if R is Noetherian (respectively, Artinian). Moreover, the set of vertices of (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, (R) has n vertices (n < 1) if and only if R has only n nonzero proper ideals. Next, we study the connectivity of (R). It is shown that (R) is a connected graph and diam(R) ≤ 3 and if (R) contains a cycle, then gr((R)) ≤ 4. Also, rings R for which the graph (R) is complete or star, are characterized, as well as rings R for which every vertex of (R) is a prime (or maximal) ideal. In Part II we shall study the diameter and coloring of annihilating-ideal graphs.
机译：设R为交换环，其中（R）的理想集合带有非零an灭子。在本文及其续篇中，我们介绍和研究R的an灭理想图，以（R）表示。它是顶点为（R）*：=（R） {（0）}的（无向）图，并且当且仅当IJ =（0）时，两个不同的顶点I和J相邻。首先，我们研究（R）的一些有限性条件。例如，表明如果R不是域，则且仅当R为Noetherian（分别为Artinian）时，（R）在顶点上具有升链条件（分别为降链条件）。此外，当R为Artinian环或可分解环时，（R）的顶点集和R的非零适当理想集的基数相同。这对于环R产生，当且仅当R仅具有n个非零的理想理想时，（R）具有n个顶点（n <1）。接下来，我们研究（R）的连通性。证明（R）是一个连通图并且diam（R）≤3，并且如果（R）包含一个循环，则gr（（R））≤4。而且，图（R）完整的环R （R）的每个顶点都是质数（或最大）理想值的环R或环R的特征。在第二部分中，我们将研究an灭理想图的直径和着色。

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