Let $R$ be a commutative ring, and let $f \in R[\![X]\!]$ be a formal power series. Sometimes (and for example, this will always be possible if $R$ is Noetherian), one may write $f$ in the form $$
f = \sum_{i=1}^n a_i u_i X^i,
$$ where $a_i \in R$ and $u_i$ is a *unit* in $R[\![X]\!]$. I call the smallest such $n$ for which such an expression is possible the *pseudodegree* of $f$.

At least, that's what I and a coauthor called it in a recent preprint. It seemed reasonable, since if one makes the same definition in the polynomial ring $R[X]$, this concept coincides with the usual notion of the *degree* of a polynomial, at least when $R$ is reduced. Also, like the degree of a polynomial, the pseudodegree of $f$ is at least as big as one less than the minimal number of generators of the ideal of $R$ that is generated by the coefficients of $f$.

But this must have a name somewhere, right? Can anyone point to an article, book, preprint or the like where this concept has been used? What else can be said about pseudodegree?