On the intersection graph of idealizations

Let R be a commutative ring with identity. The intersection graph of ideals of a ring R is an undirected simple graph denoted by Γ( 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 non-zero R -module and let R ⋉ M be the idealization of M in R . In this paper, we are interested in investigating some graph-theoretic properties of the intersection graph of idealization. We first determine the set of ideals of the Z p 2 ⋉ Z p 2 and Z pq ⋉ Z pq , for distinct prime numbers p and q . We then obtain necessary and sufficient conditions on the ring R and the module M such that Γ( R ⋉M ) is planar. In fact, we prove that Γ( R ⋉ M ) is planar if and only if R is a field and dim R ( M ) ≤ 2, or Max( R ) = {m 1 , m 2 }, m 1 ∩ m 2 = 0, m 2 M = 0 and dim R/ m2 ( M ) = 1, or M is a simple module and R has only one non-trivial ideal. MSC 2010: 13A15, 05C75