The Dedekind-Mertens lemma and the contents of polynomials

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 $\{ k \in \mathbb{N}\; \vert \; c_R(f)^{k-1} c_R(fg) = c_R(f)^{k} c_R(g) \mbox{ for all } f \in R[X] \}$, 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 $\mu_R(g)$.