Maximal sets of linearly independent vectors in a free module over a commutative ring (Q1189627)

This paper contains the following result: Theorem. Let \(R\) be a noetherian ring, and \(M\) be a free \(R\)-module of rank \(n\). Let \(\{v_ 1,v_ 2,\ldots,v_ s\}\) be a maximal set of linearly independent vectors of \(M\). Then \(s=n\). It is also pointed out, by giving an example, that the above theorem is false, if \(R\) is a commutative ring and \(n\geq 2\).