## 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.

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## Acknowledgements

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2019.06.

The authors would like to thank the referees for their comments and suggestions, which greatly helped us improve the presentation of the paper.

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Giau, L.T.N., Toan, P.T. & Vo, T.N. Dedekind–Mertens Lemma for Power Series in an Arbitrary Set of Indeterminates.
*Vietnam J. Math.* **50, **45–58 (2022). https://doi.org/10.1007/s10013-020-00466-4

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DOI: https://doi.org/10.1007/s10013-020-00466-4

### Keywords

- Content ideal
- Dedekind–Mertens lemma
- Power series ring

### Mathematics Subject Classification (2010)

- 13A15
- 13F25