TY - JOUR

T1 - Generation of summand absorbing submodules

AU - Izhakian, Zur

AU - Knebusch, Manfred

AU - Rowen, Louis

PY - 2020/8/20

Y1 - 2020/8/20

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,y V, x + y W â'x W,y 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,y V, x + y W â'x W,y 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

M3 - Article

AN - SCOPUS:85095445915

JO - Journal of Algebra and its Applications

JF - Journal of Algebra and its Applications

SN - 0219-4988

M1 - 2150201-1

ER -