Von Neumann Regular Ring - Facts

Facts

The following statements are equivalent for the ring R:

• R is von Neumann regular
• every principal left ideal is generated by an idempotent
• 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.

In a commutative von Neumann regular ring, for each element x there is a unique element y such that xyx=x and yxy=y, 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 x∈R (the unique element y such that xyx=x and yxy=y).

Also, the following are equivalent: for a commutative ring A

• is von Neumann regular.
• The spectrum of R is Hausdorff (with respect to Zariski topology).
• The constructible topology and Zariski topology for coincide.

Every semisimple ring is von Neumann regular, and a left (or right) Noetherian von Neumann regular ring is semisimple. Every von Neumann regular ring has Jacobson radical {0} and is thus semiprimitive (also called "Jacobson semi-simple").

Generalizing the above example, suppose S is some ring and M is an S-module such that every submodule of M is a direct summand of M (such modules M are called semisimple). Then the endomorphism ring End_S(M) is von Neumann regular. In particular, every semisimple ring is von Neumann regular.