# 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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## References

1. Anderson, D.D., Kang, B.G.: Content formulas for polynomials and power series and complete integral closure. J. Algebra 181, 82–94 (1996)

2. Arnold, J.T., Gilmer, R.: On the contents of polynomials. Proc. Amer. Math. Soc. 24, 556–562 (1970)

3. Bruns, W., Guerrieri, A.: The Dedekind–Mertens formula and determinantal rings. Proc. Amer. Math. Soc. 127, 657–663 (1999)

4. Corso, A., Heinzer, W., Huneke, C.: A generalized Dedekind–Mertens lemma and its converse. Trans. Amer. Math. Soc. 350, 5095–5109 (1998)

5. Dedekind, R.: ÜBer einenarithmetischen Satz von gauß. Mittheilungen der Deutschen Mathematischen Gesellschaft in Prag. Tempsky, pp. 1–11 (1892)

6. Epstein, N., Shapiro, J.: A Dedekind–Mertens theorem for power series rings. Proc. Amer. Math. Soc. 144, 917–924 (2016)

7. Gilmer, R.: Some applications of the Hilfssatz von Dedekind–Mertens. Math. Scand. 20, 240–244 (1968)

8. Gilmer, R.: Multiplicative Ideal Theory. Marcel Dekker, New York (1972)

9. Gilmer, R., Grams, A., Parker, T.: Zero divisors in power series rings. J. Reine Angew. Math. 278–279, 145–164 (1975)

10. Heinzer, W., Huneke, C.: The Dedekind–Mertens lemma and the contents of polynomials. Proc. Amer. Math. Soc. 126, 1305–1309 (1998)

11. Kaplansky, I.: Set Theory and Metric Spaces, 2nd edn. AMS Chelsea Publishing, New York (1977)

12. Mertens, F.: ÜBer einen algebraischen Satz. S.B. Akad. Wiss. Wien (2a) 101, 1560–1566 (1892)

13. Nishimura, H.: On the unique factorization theorem for formal power series. J. Math. Kyoto Univ. 7, 151–160 (1967)

14. Northcott, D.G.: A generalization of a theorem on the content of polynomials. Proc. Camb. Philos. Soc. 55, 282–288 (1959)

15. Park, M.H., Kang, B.G., Toan, P.T.: Dedekind–mertens lemma and content formulas in power series rings. J. Pure Appl. Algebra 222, 2299–2309 (2018)

16. Vo, T.N., Toan, P.T.: The power series Dedekind–Mertens number. Commun. Algebra 47, 3481–3489 (2019)

## 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

• 13A15
• 13F25