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

Let R be a commutative ring with identity and let X={Xλ}λ∈Λ be an arbitrary set (either finite or infinite) of indeterminates over R. There are three types of power series rings in the set X over R, denoted by R[[X]] _i, i = 1,2,3, respectively. In general, R[[X]] ₁⊆ R[[X]] ₂⊆ R[[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[[X]] ₃. More precisely, if g∈ R[[X]] ₃ such that the locally minimal number of special generators of A_g is k + 1, then Afk+1Ag=AfkAfg for all f∈ R[[X]] ₃. The same formula holds if f belongs to R[[X]] _i, i = 1,2, respectively. Our result is a generalization of previously known results in which X has a single indeterminate or g is a polynomial.