Powers of an ideal
= Powers of an ideal
{title2=$I^n$}
For an <ideal> $I$ in a <commutative ring> $R$, put $I^0=R$ and let $I^n$ for $n\geq1$ be the <product of ideals> with $n$ factors equal to $I$. It consists of finite sums of products of $n$ elements of $I$. These <ideals> form a descending chain $R\supseteq I\supseteq I^2\supseteq\cdots$.
Every positive power has the same <radical of an ideal> as $I$:
$$
\sqrt{I^n}=\sqrt I\qquad(n\geq1).
$$
One inclusion follows from $I^n\subseteq I$. Conversely, if $x^m\in I$, then $x^{mn}\in I^n$, giving the other inclusion. Different <ideals> can therefore define the same <radical ideal>, even when their successive powers are all distinct.