The Dedekind-Mertens lemma and the contents of polynomials

Abstract:

Let $R$ be a commutative ring, let $X$ be an indeterminate, and let $g \in R[X]$. There has been much recent work concerned with determining the Dedekind-Mertens number $\mu _R(g)$=min$\{ k \in \mathbb {N} \; | \; c_R(f)^{k-1} c_R(fg) = c_R(f)^{k} c_R(g) \mbox { for all } f \in R[X] \}$, especially on determining when $\mu _R(g)$ = $1$. In this note we introduce a universal Dedekind-Mertens number $u \mu _R(g)$, which takes into account the fact that $\mu _S(g)$ $\leq$ deg($g$) + $1$ for any ring $S$ containing $R$ as a subring, and show that $u \mu _R(g)$ behaves more predictably than $\mu _R(g)$.