$n$-Absorbing $I$-ideals

[摘要] 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 $i\in \left\{1,\ 2,\ \dots ,\ {n+1} \right\}$. 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.

[学科分类] 公共、环境与职业健康