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 
such that for all 
, there exists a 
satisfying 
. 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 
is an indeterminate, then the required
property is evidently not fulfilled.
