### Abstract

Original language | English |
---|---|

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 |

### Fingerprint

### Keywords

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

### Cite this

*Israel Journal of Mathematics*,

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

**Decompositions of modules lacking zero sums.** / Izhakian, Zur; Knebusch, Manfred; Rowen, Louis .

Research output: Contribution to journal › Article

*Israel Journal of Mathematics*, vol. 225, no. 2, pp. 503-524. https://doi.org/10.1007/s11856-018-1661-9

}

TY - JOUR

T1 - Decompositions of modules lacking zero sums

AU - Izhakian, Zur

AU - Knebusch, Manfred

AU - Rowen, Louis

N1 - The third author thanks the University of Virginia mathematics department for its hospitality.

PY - 2018/4/30

Y1 - 2018/4/30

N2 - 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.

AB - 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.

KW - semiring

KW - lacking zero sums

KW - direct sum decomposition

KW - free (semi)module

KW - projective (semi)module

KW - indecomposable

KW - semi-complement

KW - weak complement

U2 - 10.1007/s11856-018-1661-9

DO - 10.1007/s11856-018-1661-9

M3 - Article

VL - 225

SP - 503

EP - 524

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

IS - 2

ER -