. Every Boolean ring is von Neumann regular. Facts The following statements are equivalent for the ring R: R is von Neumann regular every principal left ideal is generated by an idempotent element every finitely generated left ideal is generated by an idempotent every principal left ideal is a direct summand of the left R-module R every finitely generated left ideal is a direct summand of the left R-module R every finitely generated submodule of a projective left R-module P is a direct summand of P every left R-module is flat: this is also known as R being absolutely flat, or R having weak dimension 0. every short exact sequence of left R-modules is pure exact The corresponding statements for right modules are also equivalent to R being von Neumann regular. Every von Neumann regular ring has Jacobson radical {0} and is thus semiprimitive (also called "Jacobson semi-simple"). In a commutative von Neumann regular ring, for each element x there is a unique element y such that and , so there is a canonical way to choose the "weak inverse" of x. The following statements are equivalent for the commutative ring R: R is von Neumann regular R has Krull dimension 0 and is reduced Every localization of R at a maximal ideal is a field R is a subring of a product of fields closed under taking "weak inverses" of (the unique element y such that and ). R is a V-ring. R has the right-lifting property against the ring homomorphism determined by , or said geometrically, every regular function factors through the morphism of schemes . Also, the following are equivalent: for a commutative ring A is von Neumann regular. The spectrum of A is Hausdorff (in the Zariski topology). The constructible topology and Zariski topology for Spec(A) coincide. Generalizations and specializations Special types of von Neumann regular rings include unit regular rings and strongly von Neumann regular rings and rank rings. A ring R is called unit regular if for every a in R, there is a unit u in R such that . Every semisimple ring is unit regular, and unit regular rings are directly finite rings. An ordinary von Neumann regular ring need not be directly finite. A ring R is called strongly von Neumann regular if for every a in R, there is some x in R with . The condition is left-right symmetric.