Abstract
An R-module V over a semiring R lacks zero sums (LZS) if x+y=0 implies x=y=0. More generally, we call a submodule W of V “summand absorbing” (SA) in V if ∀x,y∈V:x+y∈W⇒x∈W,y∈W. These arise in tropical algebra and modules over idempotent semirings, as well as modules over semirings of sums of squares. We explore the lattice of finite sums of SA-submodules, obtaining analogs of the Jordan–Hölder theorem, the noetherian theory, and the lattice-theoretic Krull dimension. We pay special attention to finitely generated SA-submodules, and describe their explicit generation.
Original language | English |
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Pages (from-to) | 3262-3294 |
Number of pages | 33 |
Journal | Journal of Pure and Applied Algebra |
Volume | 223 |
Issue number | 8 |
DOIs | |
State | Published - Aug 2019 |
Bibliographical note
Publisher Copyright:© 2018 Elsevier B.V.
Keywords
- Direct sum decomposition
- Indecomposable
- Lacking zero sums
- Projective (semi)module
- Semiring
- Upper bound monoid