Von Neumann regular ring

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. 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 i {displaystyle {mathfrak {i}}} 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 i {displaystyle {mathfrak {i}}} there exists an element x in i {displaystyle {mathfrak {i}}} such that a = axa. Every field (and every skew field) is von Neumann regular: for a ≠ 0 we can take x = a−1. An integral domain is von Neumann regular if and only if it is a field. Another example of a von Neumann regular ring is the ring Mn(K) of n-by-n square matrices with entries from some field K. If r is the rank of A ∈ Mn(K), then there exist invertible matrices U and V such that (where Ir is the r-by-r identity matrix). If we set X = V−1U−1, then More generally, the matrix ring over a von Neumann regular ring is again a von Neumann regular ring. The ring of affiliated operators of a finite von Neumann algebra is von Neumann regular.