Summand absorbing submodules of a module over a semiring

Zur Izhakian (Corresponding Author), Manfred Knebusch (Corresponding Author), Louis Rowen (Corresponding Author)

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Abstract

An R-module V over a semiring R lacks zero sums (LZS) if x+y=0⇒x=y=0. More generally, a submodule W of V is summand absorbing in V if ∀x,y∈V: x+y∈W⇒x∈W,y∈W. These arise in tropical algebra and modules over idempotent semirings. We explore the lattice of summand absorbing submodules of a given LZS module, especially those that are finitely generated, in terms of the lattice-theoretic Krull dimension, and describe their explicit generation.
Original languageEnglish
Pages (from-to)3262-3294
Number of pages33
JournalJournal of Pure and Applied Algebra
Volume223
Issue number8
Early online date9 Nov 2018
DOIs
Publication statusPublished - 31 Aug 2019

Keywords

  • Semiring
  • lacking zero sums
  • direct sum decomposition
  • free (semi)module
  • projective (semi)module
  • indecomposable
  • semidirect complement
  • upper bound monoid
  • weak complement
  • Upper bound monoid
  • Lacking zero sums
  • Direct sum decomposition
  • Indecomposable
  • Projective (semi)module

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