Vol. 62, No. 1, 1976

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Equational compactness and compact topologies in rings satisfying A.C.C

David K. Haley

Vol. 62 (1976), No. 1, 99–115

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.

Mathematical Subject Classification
Primary: 08A15, 08A15
Received: 21 August 1975
Published: 1 January 1976
David K. Haley