$n$-Absorbing $I$-ideals

Document Type : Original Article

Authors

1 Department of Mathematics, University of Soran, Erbil city, Kurdistan region, Iraq.

2 Department of Mathematics, University of Garmian, Kalar city, Kurdistan region, Iraq.

10.22034/kjm.2020.109814

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

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