  Kuratowski's closure-complement problem gives rise to a monoid generated by
the closure and complement operations. Consideration of this monoid yielded an
interesting classification of topological spaces, and subsequent decades saw
further exploration using other set operations. This article is an exploration
of a natural analogue in ring theory: a monoid produced by "radical" and
"annihilator" maps on the set of ideals of a ring. We succeed in characterizing
semiprime rings and commutative dual rings by their radical-annihilator
monoids, and we determine the monoids for commutative local zero-dimensional
(in the sense of Krull dimension) rings.
