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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 fR[[X]]3, we denote by Af 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 Ag 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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Correspondence to Phan Thanh Toan.

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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