On $\varphi$-1-Absorbing Prime Ideals

Abstract

In this paper, we introduce $\varphi$-1-absorbing prime ideals in commutative rings. Let $R$ be a commutative ring with a nonzero identity $1\ne 0$ and $\varphi :\mathcal{I}(R)\to \mathcal{I}(R)\cup \{\varnothing \}$ be a function where $\mathcal{I}(R)$ is the set of all ideals of $R$. A proper ideal $I$ of $R$ is called a $\varphi$-1-absorbing prime ideal if for each nonunits $x,y,z\in R$ with $xyz\in I-\varphi (I)$, then either $xy\in I$ or $z\in I$. In addition to give many properties and characterizations of $\varphi$-1-absorbing prime ideals, we also determine rings in which every proper ideal is $\varphi$-1-absorbing prime.

10 pages. arXiv admin note: text overlap with arXiv:2005.10365

References in corpus (1)

• On weakly $1$-absorbing prime ideals of commutative rings