  We show that, for a free abelian group $G$ and prime power $p^\nu$, every
direct sum decomposition of the group $G/p^\nu G$ lifts to a direct sum
decomposition of $G$. This is the key result we use to show that, if $R$ is a
commutative von Neumann regular ring, and $\mathcal{E}$ a set of idempotents in
$R$, then the projective dimension of the ideal $\mathcal{E} R$ as an
$R$-module is the same as the projective dimension of the ideal $\mathcal{EB}$,
where $\mathcal{B}$ is the boolean algebra generated by $\mathcal{E} \cup
\{1\}$. This answers a thirty year old open question of R. Wiegand. The proof
is based on gaussian elimination on an $\omega \times \omega$ matrix, with
adaptations enabling one to pass from the integers modulo $p^\nu$ to the
integers.
