Dedekind–Mertens Lemma for Power Series in an Arbitrary Set of Indeterminates

Abstract

Let R be a commutative ring with identity and let $\mathcal {X} = \{X_{\lambda }\}_{\lambda \in {\Lambda }}$ be an arbitrary set (either finite or infinite) of indeterminates over R. There are three types of power series rings in the set $\mathcal {X}$ over R, denoted by $R[[\mathcal {X}]]_{i}$, i = 1,2,3, respectively. In general, $R[[\mathcal X]]_{1} \subseteq R[[\mathcal {X}]]_{2} \subseteq R[[\mathcal {X}]]_{3}$ and the two containments can be strict. For a power series f ∈ R[[X]]₃, we denote by A_f the ideal of R generated by the coefficients of f. In this paper, we show that a Dedekind–Mertens type formula holds for power series in $R[[\mathcal {X}]]_{3}$. More precisely, if $g\in R[[\mathcal {X}]]_{3}$ such that the locally minimal number of special generators of A_g is k + 1, then $A_{f}^{k+1}A_{g} = {A_{f}^{k}} A_{fg}$ for all $f \in R[[\mathcal X]]_{3}$. The same formula holds if f belongs to $R[[\mathcal {X}]]_{i}$, i = 1,2, respectively. Our result is a generalization of previously known results in which $\mathcal X$ has a single indeterminate or g is a polynomial.