e-Prime Ideals of Commutative Rings

In this paper, all rings will be commutative with identity. We let ID(R) denote the set of all idempotents of R. In recent years, many generalizations of the concept of prime ideal have been studied such as weakly prime ideals \cite{Anderson}, S-prime ideals \cite{Hamed}, and 1-absorbing prime ideals \cite{Yassine}. In this paper, we study a new class of ideals, called e-prime ideals. Let R be a ring and let e in ID(R) be a fixed idempotent element. A proper ideal I of R with e not in I is said to be an e-prime ideal of R if whenever x and y are in R with xy in I, then ex is in I or ey is in I. Among other things, we show that I is an e-prime ideal of R if and only if R/I is an \bar{e}-domain. We also investigate e-prime ideals in direct product and polynomial rings.