V-ring (ring theory) explained

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]

Notes and References

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