In mathematics, a von Neumann regular ring is a ring R such that for every a in R there exists an x in R such that a = axa. To avoid the possible confusion with the regular rings and regular local rings of commutative algebra (which are unrelated notions), von Neumann regular rings are also called absolutely flat rings, because these rings are characterized by the fact that every left module is flat.
One may think of x as a "weak inverse" of a. In general x is not uniquely determined by a.
Von Neumann regular rings were introduced by von Neumann (1936) under the name of "regular rings", during his study of von Neumann algebras and continuous geometry.
An element a of a ring is called a von Neumann regular element if there exists an x such that a=axa. An ideal is called a (von Neumann) regular ideal if it is a von Neumann regular non-unital ring, i.e if for every element a in there exists an element x in such that a=axa.
Read more about Von Neumann Regular Ring: Examples, Facts, Generalizations and Specializations
Famous quotes containing the words von, neumann, regular and/or ring:
“Two souls dwell, alas! in my breast.”
—Johann Wolfgang Von Goethe (1749–1832)
“It means there are times when a mere scientist has gone as far as he can. When he must pause and observe respectfully while something infinitely greater assumes control.”
—Kurt Neumann (1906–1958)
“He hung out of the window a long while looking up and down the street. The world’s second metropolis. In the brick houses and the dingy lamplight and the voices of a group of boys kidding and quarreling on the steps of a house opposite, in the regular firm tread of a policeman, he felt a marching like soldiers, like a sidewheeler going up the Hudson under the Palisades, like an election parade, through long streets towards something tall white full of colonnades and stately. Metropolis.”
—John Dos Passos (1896–1970)
“The life of man is a self-evolving circle, which, from a ring imperceptibly small, rushes on all sides outwards to new and larger circles, and that without end.”
—Ralph Waldo Emerson (1803–1882)