Abstract
This paper gives necessary and sufficient conditions which guarantee that a ring have maximal left, right, or two-sided ideals. The relations of rings without maximal ideals to the Jacobson and Brown-McCoy radical are discussed. Examples of rings without maximal ideals are given to illustrate the theory. Connections of existence of maximal ideals and the Axiom of Choice in Zermelo-Fraenkel set theory are noted.
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