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von Neumann Regular Ring


A von Neumann regular ring is a ring R such that for all a in R, there exists a b in R satisfying a=aba (Jacobson 1989, p. 196).

More formally, a ring R is regular in the sense of von Neumann iff the following equivalent conditions hold.

1. Every R-module is flat.

2. R/I is a projective R-module for every finitely generated ideal I.

3. Every finitely generated right ideal is generated by an idempotent.

4. Every finitely generated right ideal is a direct summand of R.


See also

Regular Ring

This entry contributed by Margherita Barile

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References

Goodearl, K. R. Von Neumann Regular Rings, 2nd ed. Malabar, FL: Krieger, 1991.Jacobson, N. Basic Algebra II, 2nd ed. New York: W. H. Freeman, 1989.Rowen, L. "Regular Rings." In Ring Theory, Vol. 1. London: Academic Press, pp. 276-278, 1988.Weibel, C. An Introduction to Homological Algebra. New York: Cambridge University Press, p. 98, 1994.

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von Neumann Regular Ring

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

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