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Published online by Cambridge University Press: 20 November 2018
We present here an investigation of the theory of one-sided ideals in a topological ring R. One of our aims is to discuss the question of "left" properties versus "right" properties. A problem of this sort is to decide if (a) all the modular maximal right ideals of R are closed if and only if all the modular maximal left ideals of R are closed. It is shown that this is the case if R is a quasi-Q-ring, that is, if R is bicontinuously isomorphic to a dense subring of a Q-ring (for the notion of a Q-ring see (6) or §2). All normed algebras are quasi-Q-rings. Also (a) holds if R is a semisimple ring with dense socle.