In this question, I asked about commutative finite-dimensional $\Bbb C$-algebras without nilpotents. Turns out that all such algebras are isomorphic to $\Bbb C$^n.
How does this classification change if we look at commutative finite-dimensional $\Bbb Z$-algebras instead? These can also be thought of as finite-dimensional semiprime rings, or as reduced rings, which are equivalent in the commutative case.
All of the examples I've looked at so far seem to be a product of integral domains - things like $\Bbb Z$, finite fields $GF(p^n)$'s, or ring extensions such as $\Bbb Z[i]$. This leads to the following conjecture:
Is every finitely generated commutative semiprime $\Bbb Z$-algebra a direct product of integral domains?
If not, then in general, is every finite-dimensional commutative semiprime ring a direct product of something? If so, what?
EDIT: this page says that every finite-dimensional commutative semiprime ring injects into a direct product of integral domains, e.g. it is a "subdirect product". The question is now whether or not this reduces to a full direct product in the finitely generated case.