Decompositions of modules lacking zero sums

TL;DR

This paper develops a decomposition theory for modules over semirings lacking zero sums, revealing unique complements and decompositions in specific algebraic structures like max-plus semirings.

Contribution

It introduces a new decomposition framework for modules over semirings without zero sums, including uniqueness results and conditions for finitely generated projective modules.

Findings

Unique direct complements in max-plus semirings

Decomposition of finitely generated projective modules

Conditions for the existence of primitive idempotents

Abstract

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.