V-ring (ring theory)

In mathematics, a V-ring is a ring R such that every simple R-module is injective. The following three conditions are equivalent:[1]

  1. Every simple left (resp. right) R-module is injective
  2. The radical of every left (resp. right) R-module is zero
  3. Every left (resp. right) ideal of R is an intersection of maximal left (resp. right) ideals of R

A commutative ring is a V-ring if and only if it is Von Neumann regular.

References

  1. ↑ Faith, Carl (1973). Algebra: Rings, modules, and categories. Springer-Verlag. ISBN 0387055517.
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