= Strict descent of powers of a nonzero proper ideal in a Noetherian domain
{title2=$I^n\supsetneq I^{n+1}$}
Let $R$ be a <Noetherian ring> that is an <integral domain>, and let $I$ be a nonzero proper <ideal>. If $I^n=I^{n+1}$ for some $n\geq1$, the nonzero <finitely generated module> $M=I^n$ satisfies $M=IM$. Choose generators $u_1,\ldots,u_r$ and write $u_i=\sum_j a_{ij}u_j$ with $a_{ij}\in I$. The <adjugate matrix> identity implies that $\det(\operatorname{Id}_r-(a_{ij}))$ annihilates $M$. This <determinant> is $1-b$ for some $b\in I$, since every term except the constant term contains an entry from $I$.
The <determinant trick> has thus supplied $(1-b)M=0$. A nonzero element of $M$ and <multiplicative cancellation in an integral domain> force $b=1$, contradicting properness of $I$. Therefore all positive <powers of an ideal> are distinct in this setting. Since they have the same <radical of an ideal>, there are infinitely many <ideals> with radical $\sqrt I$.
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