Summand absorbing submodules of a module over a semiring

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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.




Izhakian, Z., Knebusch, M., & Rowen, L. (2019). Summand absorbing submodules of a module over a semiring. Journal of Pure and Applied Algebra, 223(8), 3262–3294.

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