  A direct sum decomposition theory is developed for direct summands (and
complements) of modules over a semiring $R$, having the property that $v+w = 0$
implies $v = 0$ and $w = 0$. Although this never occurs when $R$ is a ring, it
always does holds for free modules over the max-plus semiring and related
semirings. In such situations, the direct complement is unique, and the
decomposition is unique up to refinement. Thus, every finitely generated
projective module is a finite direct sum of summands of $R$ (assuming the mild
assumption that $1$ is a finite sum of orthogonal primitive idempotents of
$R$). Some of the results are presented more generally for weak complements and
semidirect complements. We conclude by examining the obstruction to the "upper
bound" property in this context.
