TY - JOUR
T1 - Generation of summand absorbing submodules
AU - Izhakian, Zur
AU - Knebusch, Manfred
AU - Rowen, Louis
N1 - Publisher Copyright:
© 2021 World Scientific Publishing Company.
PY - 2021/11/1
Y1 - 2021/11/1
N2 - An R-module V over a semiring R lacks zero sums (LZS) if x + y = 0 implies x = y = 0. More generally, a submodule W of V is "summand absorbing"(SA), if, for all x,yϵV, x + yϵW→xϵW,y ϵ W. These relate to tropical algebra and modules over (additively) idempotent semirings, as well as modules over semirings of sums of squares. In previous work, we have explored the lattice of SA submodules of a given LZS module, especially, those that are finitely generated, in terms of the lattice-theoretic Krull dimension. In this paper, we consider which submodules are SA and describe their explicit generation.
AB - An R-module V over a semiring R lacks zero sums (LZS) if x + y = 0 implies x = y = 0. More generally, a submodule W of V is "summand absorbing"(SA), if, for all x,yϵV, x + yϵW→xϵW,y ϵ W. These relate to tropical algebra and modules over (additively) idempotent semirings, as well as modules over semirings of sums of squares. In previous work, we have explored the lattice of SA submodules of a given LZS module, especially, those that are finitely generated, in terms of the lattice-theoretic Krull dimension. In this paper, we consider which submodules are SA and describe their explicit generation.
KW - Additive spine
KW - Direct sum decomposition
KW - Free (semi)module
KW - Halo
KW - Lacking zero sums
KW - Matrices
KW - Semigroup
KW - Semiring
KW - Summand absorbing submodule
KW - Tropical space
UR - http://www.scopus.com/inward/record.url?scp=85095445915&partnerID=8YFLogxK
U2 - 10.1142/S0219498821502017
DO - 10.1142/S0219498821502017
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AN - SCOPUS:85095445915
SN - 0219-4988
VL - 20
JO - Journal of Algebra and its Applications
JF - Journal of Algebra and its Applications
IS - 11
M1 - 2150201
ER -