A regular ring in the sense of commutative algebra is a commutative unit ring such that all its localizations at prime ideals are regular local rings. In contrast, a von Neumann regular ring is an object of noncommutative ring theory defined as a ring R such that for all a element R, there exists a b element R satisfying a = a b a. von Neumann regular rings are unrelated to regular rings (or regular local rings) in the sense of commutative algebra. For example, a polynomial ring over a field is always regular in the sense of commutative algebra, but is certainly not regular in the sense of von Neumann, since if a is an indeterminate, then the required property is evidently not fulfilled.
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