A property of independent elements (Q1063645)

The elements \(x_ 1,x_ 2,...,x_ k\) in a commutative ring are called independent if \(x_ 1r_ 1+x_ 2r_ 2+...+x_ kr_ k=0\) implies that \(r_ 1,r_ 2,...,r_ k\) belongs to \((x_ 1,x_ 2,...,x_ k)\). The paper consists of an elegant proof of the following theorem: A set of independent elements cannot be contained in a proper ideal generated by fewer elements. As a corollary one has the following (well-known) result: For any local flat morphism \(A\to B\) of Noetherian rings, emb.dim \(A\leq emb.\dim B\).