Regular rings and integral extension of a regular ring
Abstract: In this paper we show that a ring (not necessarily commutative) with identity element and without nonzero nilpotent elements is a von Neumann regular ring if every completely prime ideal is a maximal right ideal. Using this result, we show an integral extension (not necessarily commutative) without nonzero nilpotent elements of a regular ring is itself a regular ring.
- V. A. Andrunakievič and Ju. M. Rjabuhin, Rings without nilpolent elements and completely simple ideals, Dokl. Akad. Nauk SSSR 180 (1968), 9-ll=Soviet Math. Dokl. 9 (1968), 565-567. MR 37 #6320. MR 0230760 (37:6320)
- I. Kaplansky, Commutative rings, Allyn and Bacon, Boston, Mass., 1970. MR 40 #7234. MR 0254021 (40:7234)
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Keywords: Completely prime ideal, semicompletely prime ideal, m-system, multiplicatively closed system, regular ring, integral extension
Article copyright: © Copyright 1972 American Mathematical Society