In a 1962 paper S. Warner
posed the following question: for which classes of rings does each compact topological
ring therein possess a unique compact topology? There it is shown that the
class of (unital) noetherian rings has this property, the topology in question
being the Jacobson radical topology; in another paper he showed that the
class of semisimple rings enjoys the same property, this latter result drawing
heavily on I. Kaplansky’s structure theorem for compact semisimple rings. In
particular a compact topological ring with no nonzero topological nilpotents is
in possession of the only compact topology it can carry. In this report we
investigate equational compactness in the class of arbitrary (associative)
rings satisfying the ascending chain condition on left ideals (A.C.C.). Now
the theory of equational compactness has been probed in various classes of
(universal) algebras and in many cases this algebraic property characterizes the
topological-algebraic property that an algebra be a retract of a compact
topological algebra (the “Mycielski Problem”). We showed that in the class of
commutative noetherian rings equational compactness is equivalent to topological
compactness, so a strong motivating factor prompting a further investigation
was the suspicion that the sharpened result obtained in the commutative
noetherian case could be substantially generalized — yielding, first, an answer to
the Mycielski Problem in this larger class of rings and, secondly, providing
additional algebraic footholds for a further assault on Warner’s question. Here
positive answers to both questions are obtained for the class of rings with
A.C.C.
