Independence of elements in a ring and the height of the ideal they generate

We discuss three closely connected topics concerning the independence of elements in a commutative ring and the height of the ideal they generate. The first topic discussed is the relative independence of elements in a commutative ring. We start by considering a general ring and then specialize it to a local ring and in particular to and a power series ring, comparing the notion of relative independence to that of regular sequence, system of parameters and analytic independence respectively. Then, concentrating in the power series ring, we consider a condition under which power series are analytically independent. As the final topic we analyze the behavior of the height of an ideal in polynomial and power series extensions