The ring of an outer von Neumann frame in modular lattices

We prove the following theorem. Let (a1, . . . , am, c12, . . . , c1m) be a spanning von Neumann m-frame of a modular lattice L, and let (u1, . . . , un, v12, . . . , v1n) be a spanning von Neumann n-frame of the interval [0, a1]. Assume that either m ≥ 4, or L is Arguesian and m ≥ 3. Let R* denote the coordinate ring of (a1, . . . , am, c12, . . . , c1m). If n ≥ 2, then there is a ring S* such that R* is isomorphic to the ring of all n × n matrices over S*. If n ≥ 4 or L is Arguesian and n ≥ 3, then we can choose S* as the coordinate ring of (u1, . . . , un, v12, . . . , v1n).