  Hindman's Theorem says that every finite coloring of the positive natural
numbers has a monochromatic set of finite sums. Ramsey algebras, recently
introduced, are structures that satisfy an analogue of Hindman's Theorem. It is
an open problem posed by Carlson whether every Ramsey algebra has an idempotent
ultrafilter. This paper developes a general framework to study idempotent
ultrafilters. Under certain countable setting, the main result roughly says
that every nondegenerate Ramsey algebra has a nonprincipal idempotent
ultrafilter in some nontrivial countable field of sets. This amounts to a
positive result that addresses Carlson's question in some way.
