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 xR (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 EndS(M) is von Neumann regular. In particular, every semisimple ring is von Neumann regular.

Read more about this topic:  Von Neumann Regular Ring

Famous quotes containing the word facts:

    Obviously the facts are never just coming at you but are incorporated by an imagination that is formed by your previous experience. Memories of the past are not memories of facts but memories of your imaginings of the facts.
    Philip Roth (b. 1933)

    Now, what I want is, Facts. Teach these boys and girls nothing but Facts. Facts alone are wanted in life. Plant nothing else, and root out everything else. You can only form the minds of reasoning animals upon Facts: nothing else will ever be of any service to them. This is the principle on which I bring up my own children, and this is the principle on which I bring up these children. Stick to Facts, sir!
    Charles Dickens (1812–1870)

    Is it true or false that Belfast is north of London? That the galaxy is the shape of a fried egg? That Beethoven was a drunkard? That Wellington won the battle of Waterloo? There are various degrees and dimensions of success in making statements: the statements fit the facts always more or less loosely, in different ways on different occasions for different intents and purposes.
    —J.L. (John Langshaw)