Resumen de Intersection graph of idealizations

Akram Mahmoodi, Alireza Vahidi, Raoufeh Manaviyat, Roghaieh Alipour

• Let R be a commutative ring with identity. The intersection graph of ideals of a ring R is an undirected simple graph denoted by 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 be the idealization of M in R. In this paper, we investigate the interplay between the algebraic properties of and the graph-theoretic properties of . Under some conditions on the ring R and the module M, we determine the exact form of ideals of and characterize all rings R and modules M for which is a star graph. Also, we give a necessary and sufficient condition under which is uniform and then we conclude that when is a complete graph. Among other results, some graph-theoretic properties of such as domination number, connectedness and grith are obtained.