Let R be a commutative ring with identity. The intersection graph of ideals of a ring R is an undirected simple graph denoted by \(\Gamma (R)\) whose vertices are in a one-to-one correspondence with non-zero proper ideals and two distinct vertices are joined by an edge if and only if the corresponding ideals of R have a non-zero intersection. Let M be a unitary R-module and let \(R\ltimes M\) be the idealization of M in R. In this paper, we investigate the interplay between the algebraic properties of \(R\ltimes M\) and the graph-theoretic properties of \(\Gamma (R\ltimes M)\). Under some conditions on the ring R and the module M, we determine the exact form of ideals of \(R\ltimes M\) and characterize all rings R and modules M for which \(\Gamma (R\ltimes M)\) is a star graph. Also, we give a necessary and sufficient condition under which \(R\ltimes M\) is uniform and then we conclude that when \(\Gamma (R\ltimes M)\) is a complete graph. Among other results, some graph-theoretic properties of \(\Gamma (R\ltimes M)\) such as domination number, connectedness and grith are obtained.

令R为具有恒等式的交换环。环R的理想交图是一个无向简单图，用\(\Gamma (R)\)表示，其顶点与非零本真理想一一对应，两个不同的顶点由一个边当且仅当R的相应理想具有非零交点。令M为酉R模，令\(R\ltimes M\)为M在R中的理想化。在本文中，我们研究了\(R\ltimes M\)的代数性质与\(\伽玛 (R\ltimes M)\)。在环R和模M的某些条件下，我们确定\(R\ltimes M\)理想的确切形式，并表征所有环R和模M，其中\(\Gamma (R\ltimes M)\)是星图。此外，我们给出了\(R\ltimes M\)是均匀的充分必要条件，然后我们得出结论，当\(\Gamma (R\ltimes M)\)是一个完全图。在其他结果中，获得了\(\Gamma (R\ltimes M)\)的一些图论属性，例如支配数、连通性和粒度。