Abstract
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.
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Mahmoodi, A., Vahidi, A., Manaviyat, R. et al. Intersection graph of idealizations. Collect. Math. (2023). https://doi.org/10.1007/s13348-023-00407-7
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DOI: https://doi.org/10.1007/s13348-023-00407-7