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 .