A Dedekind-Mertens theorem for power series rings (Q2790166)

Let \(R\) be a commutative ring. For \(f\in R[X]\) a polynomial, the content of \(f\) written \(c(f)\) is the ideal of \(R\) generated by the coefficients of \(f\). The Dedekind-Mertens lemma says that for \(f,g\in R[X]\), we have \(c(f)^kc(g)=c(f)c(fg)\) for some positive integer \(k\). For a formal power series \(f\in R[[X]]\), we may define the content of \(f\) to be the ideal generated by the coefficients. In this paper, the authors extend Dedekind-Mertens lemma for formal power series when \(R\) is Noetherian. Many examples are given.