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

TL;DR

This paper introduces and studies weakly 1-absorbing prime ideals in commutative rings, generalizing weakly prime ideals, with characterizations, properties, and examples in various ring classes.

Contribution

It defines weakly 1-absorbing prime ideals, explores their properties, and characterizes them in PIDs and Dedekind domains, extending the theory of prime ideals.

Findings

Weakly 1-absorbing prime ideals generalize weakly prime ideals.

If such an ideal is not 1-absorbing prime, then its cube is zero.

Characterizations are provided for PIDs and Dedekind domains.

Abstract

Let $R$ be a commutative ring with identity. In this paper, we introduce the concept of weakly $1$-absorbing prime ideals which is a generalization of weakly prime ideals. A proper ideal $I$ of $R$ is called weakly $1$-absorbing prime if for all nonunit elements $a,b,c \in R$ such that $0\neq abc \in I$, then either $ab \in I$ or $c \in I$. A number of results concerning weakly $1$-absorbing prime ideals and examples of weakly $1$-absorbing prime ideals are given. It is proved that if $I$ is a weakly $1$-absorbing prime ideal of a ring $R$ and $0 \neq I_1I_2I_3 \subseteq I$ for some ideals $I_1, I_2, I_3$ of $R$ such that $I$ is free triple-zero with respect to $I_1I_2I_3$, then $ I_1I_2 \subseteq I$ or $I_3\subseteq I$. Among other things, it is shown that if $I$ is a weakly $1$-absorbing prime ideal of $R$ that is not $1$-absorbing prime, then $I^3 = 0$. Moreover, weakly $1$-absorbing prime ideals of PID's and Dedekind domains are characterized. Finally, we investigate commutative rings with the property that all proper ideals are weakly $1$-absorbing primes.