NOTES ON 1-ABSORBING PRIME IDEALS

Abstract

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 is an element of I for some nonunits x, y, z is an element of R, then xy is an element of I or z is an element of I. It is well known that prime ideal double right arrow 1-absorbing prime ideal double right arrow primary ideal double right arrow 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.