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Regular Ring


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 in R, there exists a b in R satisfying a=aba. 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.


See also

Regular Local Ring, Ring, von Neumann Regular Ring

This entry contributed by Margherita Barile

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References

Bruns, W. and Herzog, J. Cohen-Macaulay Rings, 2nd ed. Cambridge, England: Cambridge University Press, 1998.

Referenced on Wolfram|Alpha

Regular Ring

Cite this as:

Barile, Margherita. "Regular Ring." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/RegularRing.html

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