Perspectivity in complemented modular lattices and regular rings

Abstract

Based on an analogue for systems of partial isomorphisms between lower sections in a complemented modular lattice we prove that principal right ideals $aR \cong bR$ in a (von Neumann) regular ring $R$ are perspective if $aR \cap bR$ is of finite height in $L(R)$. This is applied to derive, for existence-varieties $\mathcal{V}$ of regular rings, equivalence of unit-regularity and direct finiteness, both conceived as a property shared by all members of $\mathcal{V}$.