Bases in semimodules over commutative semirings

In this paper, the bases of a semimodule over a commutative semiring R are investigated. Some properties and characterizations of the bases are discussed and some equivalent conditions for a basis to be a free basis in a finitely generated free semimodule over R are given. The different possible cardinalities for a basis in a finitely generated free semimodule over R are considered and some equivalent descriptions are obtained for a commutative semiring R satisfying the property that any two bases for a finitely generated free R-semimodule have the same cardinality. Partial results obtained in the paper develop and generalize the corresponding results for commutative join-semirings and for commutative zerosumfree semirings.