ON STRONGLY r-IDEALS ON COMMUTATIVE RINGS AND SOME RELATED GRAPHS

Abstract

A proper ideal I of a ring R is called an r-ideal if, whenever x,y ∈ I with xy ∈ I, we have x ∈ I or y ∈ Z(R) [15]. In this article, we are interested in a subclass of the class of r-ideals which we call the class of strongly r-ideals. A proper ideal I of a ring R is called a strongly r-ideal if, whenever x,y ∈ I with xy ∈ I, we have x ∈ I or y ∈ Z(I). First, we give a basic study of this new concept which includes, among others, characterizations, properties and examples. After that, we use the introduced concept to characterize rings for which the diameter of the zero-divisor graph is less than or equal to two, rings for which the annihilator graph is complete, and rings for which the zero-annihilator graph is empty.