Notes On Some Basis Over Semimodules

The paper investigates the characterizations of strongly linear independent set-s and strongly linear independent basis in finitely generated semimodules over commutative semirings and the cardinality of free basis in finitely generated semimodules over stably finite semirings.It first discuss the characterizations of strongly linear independent sets and strongly linear independent basis,which is then used to study the relationships between the strongly lin-ear independent sets and GM-rank of semimodules in finitely generated semimodules over commutative semirings.It finally gives an equivalent description that for a basis to be a free basis and each basis has the same number of elements in finitely generated semimodules over stably finite semirings.