On projective modules over directly finite regular rings satisfying the comparability axiom (Q1064386)

Let R be a directly finite von Neumann regular ring satisfying the comparability axiom, i.e. for all x,y\(\in R\) there exists a monomorphism xR\(\to yR\) or a monomorphism yR\(\to xR\) [see \textit{K. R. Goodearl}, Von Neumann regular rings (1979; Zbl 0411.16007)]. In the present note the author describes the structure of the directly finite projective R- modules and, as a corollary, proves that the direct sum of two directly finite projective R-modules is again directly finite. If, in addition, R is simple, directly finite projective R-modules have already been characterized by \textit{J. Kado} [Osaka J. Math. 16, 405-412 (1979; Zbl 0415.16013)].