Let $\mathfrak{a}$ be a monomial ideal in a polynomial algebra over some commutative ring $R$. If $R$ is reduced, then the radical $\sqrt{\mathfrak{a}}$ of $\mathfrak{a}$ is again a monomial ideal, and if $\mathfrak{a}$ is moreover finitely generated then so is $\sqrt{\mathfrak{a}}$. If $R$ is not reduced, then the nilradical of $R$ is contained in $\sqrt{\mathfrak{a}}$ and may make the latter somewhat less easy to handle. However, if $\mathfrak{a}$ and the nilradical of $R$ are finitely generated then so is $\sqrt{\mathfrak{a}}$.

This leads to the following question:

Is there a nice description of rings $R$ such that the nilradical of $R$ is finitely generated?

Or in more geometric terms:

Is there a nice description of schemes $X$ such that the associated reduced scheme $X_{{\rm red}}$ is locally of finite presentation over $X$?

goodcase includes all Noetherian rings. My impression is that in general in commutative algebra if you can show something for all Noetherian rings you are pretty happy...*unless* you have a specific class of non-Noetherian rings in mind. (As for me, the non-Noetherian rings I like are either domains or rings of functions, both of which are reduced.) Do you? $\endgroup$ – Pete L. Clark Jul 15 '11 at 23:42