Decompositions of modules lacking zero sums

A module over a semiring lacks zero sums (LZS) if it has the property that v +w = 0 implies v = 0 and w = 0. While modules over a ring never lack zero sums, this property always holds for modules over an idempotent semiring and related semirings, so arises for example in tropical mathematics. A direct sum decomposition theory is developed for direct summands (and complements) of LZS modules: The direct complement is unique, and the decomposition is unique up to refinement. Thus, every finitely generated “strongly 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). This leads to an analog of the socle of “upper bound” modules. Some of the results are presented more generally for weak complements and semi-complements. We conclude by examining the obstruction to the “upper bound” property in this context.