Notes on 1-Absorbing Prime Ideals

Let R be a commutative ring with a nonzero identity. A proper ideal I of R is said to be a 1-absorbing prime ideal if xyz ∈ I for some nonunits x, y, z ∈ R, then xy ∈ I or z ∈ I. It is well known that prime ideal ⇒ 1-absorbing prime ideal ⇒ primary ideal ⇒ semi-primary ideal, that is, the class of 1-absorbing prime ideals comes between the classes of prime ideals and primary ideals. Also, the above right arrows are not reversible. In this article, we characterize rings over which every 1-absorbing prime ideal is prime and every primary ideal is 1-absorbing prime. Also, by comparing 1-absorbing prime ideals and other some classical ideals such as 2-absorbing ideals and semi-primary ideals, we characterize Noetherian divided rings and von Neumann regular rings.