Affiliation:
1. Department of Mathematics, Faculty of Sciences, King Khalid University, Abha, Saudi Arabia
2. Mathematics Department, Pluridisciplinary faculty, Mohammed First University, Selouane, Nador, Morocco
3. Department of Mathematics, Faculty of Science, University Moulay Ismail Meknes, Zitoune, Morocco
Abstract
A proper ideal I of a ring R is called an r-ideal if, whenever x, y ? R with
xy ? I, we have x ? I or y ? Z(R) [R. Mohamadian, r-ideals in commutative
rings, Turkish J. Math. 39(5) (2015),733-749]. 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 ? R 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.
Publisher
National Library of Serbia