  The first part of the paper is a brief overview of Hindman's finite sums
theorem, its prehistory and a few of its further generalizations, and a modern
technique used in proving these and similar results, which is based on
idempotent ultrafilters in ultrafilter extensions of semigroups. The second,
main part of the paper is devoted to the topologizability problem of a wide
class of algebraic structures called polyrings; this class includes Abelian
groups, rings, modules, algebras over a ring, differential rings, and others.
We show that the Zariski topology of such an algebra is always non-discrete.
Actually, a much stronger fact holds: if $K$ is an infinite polyring, $n$ a
natural number, and a map $F$ of $K^n$ into $K$ is defined by a term in $n$
variables, then $F$ is a closed nowhere dense subset of the space $K^{n+1}$
with its Zariski topology. In particular, $K^n$ is a closed nowhere dense
subset of $K^{n+1}$. The proof essentially uses a multidimensional version of
Hindman's finite sums theorem established by Bergelson and Hindman. The third
part of the paper lists several problems concerning topologization of various
algebraic structures, their Zariski topologies, and related questions.
