2-AB rings and 2-absorbing rings

Abstract

Let R be a commutative ring with identity. The notion of 2-absorbing ideal was introduced by A. Badawi (Bull Aust Math Soc 75(3):417–429, 2007), as a generalization of prime ideal. A proper ideal I of R is called a 2-absorbing ideal of R if whenever a, b, $c\in R$ with $abc\in I$, then $ab\in I$ or $ac\in I$ or $bc\in I$. A commutative ring R is called a 2-absorbing ring if every nonzero proper ideal of R is 2-absorbing. It is clear that every prime ideal is a 2-absorbing ideal but the converse is not true in general. D. Bennis and B. Fahid introduced a new class of rings in which every 2-absorbing ideal is prime. They called such a ring, a 2-AB ring (Bennis in Beitr Algebra Geom 59(2):391–396, 2018). They showed that if R is a commutative ring such that the prime ideals of R are comparable and $P^2=P$, for every prime ideal P of R then R is 2-AB and they asked if $P^2=P$ for every prime ideal in a 2-AB ring. In this paper, we give a positive answer to this question in various classes of rings and we investigate the transfer of the 2-AB (respectively 2-absorbing) property to some ring extensions.