Abstract
In 1947 Kaplansky proved that if A is a compact ring with Jacobson radical J, then A⧸ J is isomorphic and homeomorphic to a Cartesian product of matrix rings over finite fields. That result is used to give additional structure theorems for compact rings. In particular those commutative compact rings with identity for which 2 is a unit are characterized. Moreover, an analogue of a result of Andrunakievič and Rjabuhin is obtained, namely, that if A is a compact ring with identity such that A satisfies the ascending chain condition on closed ideals, then A is isomorphic and homeomorphic to a subdirect sum of compact local rings without zero divisors if and only if A has no nonzero algebraic nilpotent elements.
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