The radical of an n-absorbing ideal

TL;DR

This paper proves that in a commutative ring with unity, the radical of an n-absorbing ideal raised to the n-th power is contained within the ideal itself, revealing a key algebraic property.

Contribution

It establishes a new inclusion relation between the radical of an n-absorbing ideal and the ideal, generalizing known properties of ideals in ring theory.

Findings

For any n-absorbing ideal I, (\sqrt{I})^n extless{} I

The result holds in commutative rings with unity

Provides insight into the structure of n-absorbing ideals

Abstract

In this note we show that in a commutative ring $R$ with unity, for any $n > 0$, if $I$ is an $n$-absorbing ideal of $R$, then $(\sqrt{I})^{n} \subseteq I$.