On the rank of semimodules over semirings
Let \((_SH,+)\) be a left semimodule over a semiring \((S,+,\cdot)\) with commutative addition. By a basis \(U\) of \((_SH,+)\) we shall mean a generating subset of \(H\) such that every element of \(H\) is a unique linear combination of \(U\). A semiring \(S\) is said to have property \(R(m,n)\) for natural numbers \(m\), \(n\) iff there is a left semimodule \((_SH,+)\) over \(S\) which has two bases \(U\) and \(V\) with \(\text{card }U=m\) and \(\text{card }V=n\). In this paper the authors investigate conditions on a semiring \(S\) having the property \(R(m,n)\), where \(m\neq n\). This includes and generalizes corresponding results on rings, in particular one due to Paul Dubreil.
scientific article; zbMATH DE number 869946