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:
“We are on a mission: we are called to the cultivation of the earth.”
—Novalis [Friedrich Von Hardenberg] (1772–1801)
“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)
“[I]n our country economy, letter writing is an hors d’oeuvre. It is no part of the regular routine of the day.”
—Thomas Jefferson (1743–1826)
“Full fathom five thy father lies,
Of his bones are coral made;
Those are pearls that were his eyes;
Nothing of him that doth fade,
But doth suffer a sea-change
Into something rich and strange.
Sea-nymphs hourly ring his knell:
Ding-dong.
Hark! Now I hear them—ding-dong bell.”
—William Shakespeare (1564–1616)