A von Neumann regular ring is a ring 
such that for all 
, there exists a 
satisfying 
(Jacobson 1989, p. 196).
More formally, a ring 
is regular in the sense of von Neumann iff
the following equivalent conditions hold.
1. Every 
-module
is flat.
2. 
is a projective 
-module for every finitely generated ideal 
.
3. Every finitely generated right ideal is generated by an idempotent.
4. Every finitely generated right ideal is a direct summand of 
.
