One may think of x as a "weak inverse" of the element a; in general x is not uniquely determined by a.
is called a (von Neumann) regular ideal if for every element a in
[1] An integral domain is von Neumann regular if and only if it is a field.
Another important class of examples of von Neumann regular rings are the rings Mn(K) of n-by-n square matrices with entries from some field K. If r is the rank of A ∈ Mn(K), Gaussian elimination gives invertible matrices U and V such that (where Ir is the r-by-r identity matrix).
[3] Generalizing the above examples, 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).
An ordinary von Neumann regular ring need not be directly finite.
A ring R is called strongly von Neumann regular if for every a in R, there is some x in R with a = aax.
In some sense, this more closely mimics the properties of commutative von Neumann regular rings, which are subdirect products of fields.