= Domain with a nonzero idempotent maximal ideal
{title2=$0\ne\mathfrak m=\mathfrak m^2$}
Let
$$
R=\bigcup_{n\geq0}\mathbb Q[u_n],\qquad u_n=t^{1/2^n},\qquad u_n=u_{n+1}^2.
$$
This <ring> consists of finite sums of rational multiples of nonnegative <dyadic rational> powers of $t$, interpreted as <functions> on $t>0$. Every pair of elements lies in one <polynomial ring> $\mathbb Q[u_n]$, so $R$ is an <integral domain>. The constant term defines a <surjective> <ring homomorphism> $R\to\mathbb Q$. Its <kernel of a ring homomorphism> $\mathfrak m$ is nonzero and is a <maximal ideal>, by the <maximal ideal quotient criterion>.
Every $f\in\mathfrak m$ has the form $u_ng(u_n)$ for some $n$ and some <polynomial> $g$. The identity
$$
f=u_{n+1}\bigl(u_{n+1}g(u_{n+1}^2)\bigr)
$$
expresses it as a product of two elements of $\mathfrak m$. Therefore $\mathfrak m=\mathfrak m^2$, and $\mathfrak m^n=\mathfrak m$ for all $n\geq1$. This shows that strict descent of <powers of an ideal> can fail in an <integral domain>, even for a nonzero <maximal ideal>. By the <determinant trick>, such an <ideal> cannot be finitely generated, so this <ring> is not a <Noetherian ring>.
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