Radical-depended graph of a commutative ring

Abstract

Let R be a commutative ring with identity and √ I be the radical of an ideal I of R. We introduce the radical-depended graph GI (R) whose vertex set is {x ∈ R \ √ I | xy ∈ I for some y ∈ R \ √ I} and distinct vertices x and y are adjacent if and only if xy ∈ I. In this paper, several properties of GI (R) are investigated and some results on the parameters of this graph are given. It follows that if I is a quasi primary ideal, then GI (R) = ∅. It is shown that if I is a 2-absorbing ideal of R which is not quasi primary, then GI (R) is the complete bipartite graph K1,1 or Km,n for some m, n ≥ 2. Moreover, it is proved that GI (R) is a connected graph with diameter at most 3, and if it has a cycle, then its girth is at most 4. Also, it is shown that if R is a Noetherian ring, then the clique number of GI (R) is equal to | Min(R/I)| for every ideal I of R.