Let R be a ring with unity and I(R)~* be the set of all nontrivial left ideals of R. The intersection graph of ideals of R, denoted by G(R), is a graph with the vertex set I(R)~* and two distinct vertices I and J are adjacent if and only if I ∩ J ≠ 0. In this paper, we study some connections between the graph-theoretic properties of this graph and some algebraic properties of rings. We characterize all rings whose intersection graphs of ideals are not connected. Also we determine all rings whose clique number of the intersection graphs of ideals is finite. Among other results, it is shown that for a ring R, if the clique number of G(R) is finite, then the chromatic number is finite and if R is a reduced ring, then both are equal.
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机译：设R为一个具有1的环，而I（R）〜*为R的所有非平凡左理想的集合。由G（R）表示的R的理想交集图是顶点集I（R ）〜*且当且仅当I are J≠0时，两个不同的顶点I和J相邻。在本文中，我们研究了该图的图论性质与环的某些代数性质之间的一些联系。我们表征所有未连接理想相交图的环。我们还要确定所有理想环的交点图的团数是有限的环。除其他结果外，表明对于环R，如果G（R）的集团数是有限的，那么色数是有限的，如果R是一个还原环，则两者是相等的。
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