Projective dimension is a lattice invariant

We show that, for a free abelian group G and prime power pν, every direct sum decomposition of the group G/pνG lifts to a direct sum decomposition of G. This is the key result we use to show that, for R a commutative von Neumann regular ring, and image a set of idempotents in R, then the projective dimension of the ideal image as an R-module the same as the projective dimension of the ideal image as a image-module, where image is the boolean algebra generated by image. This answers a 30 year old open question of R. Wiegand.