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On von Neumann regular rings, III

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Abstract

This note is a natural sequel to [8] and [9]. Further characteristic properties of arbitrary von Neumann regular rings and strongly regular rings are given in terms of annihilators and simple modules. A prime ring with certain annihilator conditions is shown to be primitive (this is related to the following problem ofKaplansky: Are prime regular rings primitive?). Necessary and sufficient conditions for leftq-rings to be regular are also considered: For example, a leftq-ring is regular iff every simple rightA-module is flat. A sufficient condition is given for a leftqc-ring to be a uniserial, strongly left and strongly rightqc, left and rightq-ring. One of the main results ofJain, Mohamed andSingh onq-rings [5, Theorem 2.13] is generalised. Finally, it is shown that a prime left continuous ring either has zero socle or is primitive, left self-injective regular.

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    Yue Chi Ming, R.: On von Neumann regular rings, II. Math. Scand.39, 167–170 (1976).

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Yue Chi Ming, R. On von Neumann regular rings, III. Monatshefte für Mathematik 86, 251–257 (1978). https://doi.org/10.1007/BF01659723

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Keywords

  • Characteristic Property
  • Prime Ring
  • Simple Module
  • Regular Ring
  • Continuous Ring