Free sets and free subsemimodules in a semimodule

In this paper, we investigate the free sets and the free subsemimodules in a semimodule over a commutative semiring S. First, we discuss some properties of the free sets and give a sufficient condition for a nonempty finite set to be free in a finitely generated free S-semimodule and obtain a relati...

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Bibliographic Details
Published in:Linear algebra and its applications 2016-05, Vol.496, p.527-548
Main Author: Tan, Yi-Jia
Format: Article
Language:English
Subjects:
Free set
Free subsemimodule
Rank of semimodule
Semimodule
Semiring
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Summary:In this paper, we investigate the free sets and the free subsemimodules in a semimodule over a commutative semiring S. First, we discuss some properties of the free sets and give a sufficient condition for a nonempty finite set to be free in a finitely generated free S-semimodule and obtain a relation between free set and linear independent set in an S-semimodule. Then we consider the free subsemimodules and prove that the rank of any free subsemimodule of a finitely generated S-semimodule M does not exceed that of M. Also, we give some equivalent descriptions for a commutative semiring S to have the property that all nonzero subsemimodules of any finitely generated free S-semimodule are free. Partial results obtained in the paper develop and generalize the corresponding results for modules over rings and linear spaces over fields.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2016.02.006