$n$-Absorbing $I$-ideals

Abstract

Let $R$ be a commutative ring with identity, let $ I $ be a proper ideal of $ R $, and let $ n ge 1 $ be a positive integer. In this paper, we introduce a class of ideals that is closely related to the class of $I$-prime ideals. A proper ideal $P$ of $R$ is called an {itshape $n$-absorbing $I$-ideal} if $a_1, a_2, dots , a_{n+1} in R$ with $a_1 a_2 dots a_{n+1} in P-IP$, then $a_1 a_2 dots a_{i-1} a_{i+1} dots a_{n+1} in P$ for some $iin left{1, 2, dots , {n+1} right}$. Among many results, we show that every proper ideal of a ring $R$ is an {itshape $n$-absorbing $I$-ideal} if and only if every quotient of $ R$ is a product of $(n+1)$-fields.