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

A commutative ring is a V-ring if and only if it is Von Neumann regular.[2]

  1. ^ Faith, Carl (1973). Algebra: Rings, modules, and categories. Springer-Verlag. ISBN 978-0387055510. Retrieved 24 October 2015.
  2. ^ Michler, G.O.; Villamayor, O.E. (April 1973). "On rings whose simple modules are injective". Journal of Algebra. 25 (1): 185–201. doi:10.1016/0021-8693(73)90088-4. hdl:20.500.12110/paper_00218693_v25_n1_p185_Michler.