  In this paper we establish a very close link (in terms of von Neumann's
coordinatization) between regular modules introduced by Zelmanowitz, on one
hand, and von Neumann regular rings, on the other hand: we prove that the
lattice $\mathcal{L}^{fg}(M)$ of all finitely generated submodules of a
finitely generated regular module $M$, over an arbitrary ring, can be
coordinatized as the lattice of all principal right ideals of some von Neumann
regular ring $S$.
