### Abstract

A module over a semiring lacks zero sums (LZS) if it has the property thatv + w = 0 implies v = 0 and w = 0. While modules over a ring never lack zero sums, thisproperty always holds for modules over an idempotent semiring and related semirings, soarises 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 torefinement. Thus, every finitely generated “strongly projective” module is a finite directsum of summands of R (assuming the mild assumption that 1 is a finite sum of orthogonalprimitive 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 semicomplements.We conclude by examining the obstruction to the “upper bound” propertyin this context.

Original language | English |
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Pages (from-to) | 503-524 |

Number of pages | 22 |

Journal | Israel Journal of Mathematics |

Volume | 225 |

Issue number | 2 |

Early online date | 11 Apr 2018 |

DOIs | |

Publication status | Published - 30 Apr 2018 |

### Keywords

- semiring
- lacking zero sums
- direct sum decomposition
- free (semi)module
- projective (semi)module
- indecomposable
- semi-complement
- weak complement

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## Profiles

### Zur Izhakian

- School of Natural & Computing Sciences, Mathematical Science - Reader
- Mathematical Sciences (Research Theme)

Person: Academic

## Cite this

Izhakian, Z., Knebusch, M., & Rowen, L. (2018). Decompositions of modules lacking zero sums.

*Israel Journal of Mathematics*,*225*(2), 503-524. https://doi.org/10.1007/s11856-018-1661-9