A Solomon-Tits Theorem for Rings

An analogue of the Tits building is defined and studied for commutative rings. We prove a Solomon-Tits theorem when R either satisfies a stable range condition, or is the ring of S-integers of a global field. We then define an analogue of the Steinberg module of R and study it both as a Z-module and as a representation. We find the rank of Steinberg when R is a finite ring, and compute the length of St(2,R)?Q as a GL(2,R)-representation when R is uniserial. As an application of these results, we produce a lower bound for the rank of the top-dimensional cohomology of principal congruence subgroups of nonprime level.