The Semimodule Type and Least Cardinality of a Free Semimodule over Semiring

The ranks of free semimodules and semimodule types of a semiring [Formula: see text] are investigated. We show that a given semiring can admit only certain characteristic types of finitely generated free semimodules, and semirings are classified according to their semimodule types. In addition, we discuss the least cardinality of a finitely generated free [Formula: see text]-semimodule and prove that if [Formula: see text] has a semimodule of type [Formula: see text], then every [Formula: see text]-semimodule with free basis of cardinality [Formula: see text] ([Formula: see text]) has no generating set of cardinality less than [Formula: see text].