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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 R n x n , the n x n matrix ring over ring R, is unit regular, then so is the ring R. It is shown that if R 2 2 is finitedoi:10.2140/pjm.1977.69.73 fatcat:oytpdflqbvhrjmv5blmjla4ioy