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

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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