Ideal containment vs. powers

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.