P-Ideals and PMP-Ideals in Commutative Rings

Recently, P-ideals have been studied in C(X) by some authors. In this article we investigate P-ideals and a new concept PMP-ideal in commutative rings. We show that I is a P-ideal (resp., PMP-ideal) in R if and only if every prime ideal of R which does not contain I is a maximal (resp., minimal prime) ideal of R. Also, we characterize the largest P-ideals (resp., PMP-ideals) in commutative rings and in C(X) as well. Furthermore, we study relations between these ideals and other ideals, such as prime, maximal, pure and von Neumann regular ideals and we find that in a reduced ring P-ideals and von Neumann regular ideals coincide. Finally, we prove that C(X) is a von Neumann regular ring if and only if all of its pure ideals are P-ideals.