A von Neumann regular ring is a ring R such that for all a element R, there exists a b element R satisfying a = a b a. More formally, a ring R is regular in the sense of von Neumann iff the following equivalent conditions hold. 1. Every R-module is flat. 2.R/I is a projective R-module for every finitely generated ideal I. 3. Every finitely generated right ideal is generated by an idempotent. 4. Every finitely generated right ideal is a direct summand of R.