Decompositions of modules lacking zero sums

Zur Izhakian, Manfred Knebusch, Louis Rowen

Research output: Contribution to journalArticle

6 Citations (Scopus)
5 Downloads (Pure)

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 languageEnglish
Pages (from-to)503-524
Number of pages22
JournalIsrael Journal of Mathematics
Volume225
Issue number2
Early online date11 Apr 2018
DOIs
Publication statusPublished - 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