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 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 MM does not exceed that of MM. 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.