$\delta (0)$-Ideals of Commutative Rings

Year 2024, Volume: 36 Issue: 36, 16 - 28, 12.07.2024

Mohamed Chhiti Bayram Ali Ersoy , Khalid Kaıba , Ünsal Tekir

https://doi.org/10.24330/ieja.1438744

Abstract

Let $R$ be a commutative ring with nonzero identity, let $\I (R)$ be the set of all
ideals of $R$ and $\delta : \I (R)\rightarrow\I (R) $ be a function. Then $\delta$ is called an expansion function of ideals of $R$ if whenever $L, I, J$ are ideals of $R$ with $J \subseteq I$, we have $L \subseteq\delta(L)$ and $\delta(J)\subseteq\delta(I)$. In this paper, we present the concept of $\dt$-ideals in commutative rings. A proper ideal $I$ of $R$ is called a $\dt$-ideal if whenever $a$, $b$ $\in R$ with $ab\in I$ and $a\notin \delta (0)$, we have $b\in I$.
Our purpose is to extend the concept of $n$-ideals to $\dt$-ideals of commutative
rings. Then we investigate the basic properties of $\dt$-ideals and also, we
give many examples about $\dt$-ideals.

Keywords

Prime ideal, $\delta$-primary ideal, $n$-ideal, $\dt$-ideal, trivial ring extension

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