Rings in which every 2-absorbing primary ideal is primary

Abstract

Let R be a commutative ring with a nonzero identity. Badawi et al. (Bull Korean Math Soc 51(4):1163–1173, 2014) defined the concept of 2-absorbing primary ideal as follows: a proper ideal P of R is said to be a 2-absorbing primary ideal if whenever $$xyz\in P$$ for some $$x,y,z\in R,$$ then either $$xy\in P$$ or $$xz\in \sqrt{P}$$ or $$yz\in \sqrt{P}.$$ It is clear that every primary ideal is also a 2-absorbing primary ideal. The author in this paper is to study rings in which every 2-absorbing primary ideal is primary. A ring R is said to be $$2-ABP$$ -ring if every 2-absorbing primary ideal is primary.