Several new properties are
derived for von Neumann finite rings. A comparison is made of the properties of von
Neumann finite regular rings and unit regular rings, and necessary and sufficient
conditions are given for a matrix ring over a regular ring to be respectively von
Neumann finite or unit regular. The converse of a theorem of Henriksen is proven,
namely that if Rn×n, the n×n matrix ring over ring R, is unit regular, then so is the
ring R. It is shown that if R2×2 is finite regular then a ∈ R is unit regular if and only
if there is x ∈ R such that R = aR + x(a0), where a0 denotes the right annihilator of
a in R.
