# Ideal containment vs. powers

## TL;DR

This paper investigates a property of ideals in commutative rings related to their powers, introduces the concept of big ideals, and characterizes rings where all ideals are Ratliff-Rush ideals, with results on regular rings and ideals without proper reductions.

## Contribution

It introduces the notion of big ideals and characterizes Noetherian domains where all ideals are Ratliff-Rush, connecting ideal properties with ring structures.

## Key findings

- Noetherian domain satisfies the property iff every ideal is Ratliff-Rush
- Ideals with no proper reduction are big ideals
- Maximal ideals in regular rings are big

## Abstract

Let $R$ be a commutative ring with identity. In this note, we study the property: If $ I \subsetneqq J$ are ideals in $R$, then $ I^n \subsetneqq J^n$ for all $ n\geq 1$. We define the notion of a big ideal (Definition 1.2). It is noted that the property has close relationship with the notions of reduction of an ideal and Ratliff-Rush ideal [7]. Apart from other results, it is proved that a Noetherian domain satifies the property if and only if every ideal in $R$ is a Ratliff-Rush ideal. We also prove that ideals having no proper reduction are big ideals, and maximal ideals in regular rings are big.