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IDEAL CHAINS IN RESIDUALLY FINITE DEDEKIND DOMAINS

Part of: Algebraic number theory: global fields Sequences and sets

Published online by Cambridge University Press:  28 November 2018

YU-JIE WANG Open the ORCID record for YU-JIE WANG [Opens in a new window] ,
YI-JING HU Open the ORCID record for YI-JING HU [Opens in a new window]  and
CHUN-GANG JI Open the ORCID record for CHUN-GANG JI [Opens in a new window]
YU-JIE WANG
Affiliation:
School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, PR China email [email protected]
YI-JING HU
Affiliation:
School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, PR China email [email protected]
CHUN-GANG JI*
Affiliation:
School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, PR China email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

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Let $\mathfrak{D}$ be a residually finite Dedekind domain and let $\mathfrak{n}$ be a nonzero ideal of $\mathfrak{D}$. We consider counting problems for the ideal chains in $\mathfrak{D}/\mathfrak{n}$. By using the Cauchy–Frobenius–Burnside lemma, we also obtain some further extensions of Menon’s identity.

Keywords

residually finite Dedekind domainMenon’s identitygroup action

MSC classification

Primary: 11R04: Algebraic numbers; rings of algebraic integers
Secondary: 11B13: Additive bases, including sumsets
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 99 , Issue 1 , February 2019 , pp. 56 - 67
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

This work was partially supported by the Grant No. 11471162 from NNSF of China and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20133207110012).

References

Li, Y. and Kim, D., ‘A Menon-type identity with many tuples of group of units in residually finite Dedekind domains’, J. Number Theory 175 (2017), 42–50.Google Scholar
Li, Y. and Kim, D., ‘Menon-type identities derived from actions of subgroups of general linear groups’, J. Number Theory 179 (2017), 97–112.Google Scholar
Menon, P. K., ‘On the sum ∑(a - 1, n)[(a, n) = 1]’, J. Indian Math. Soc. 29 (1965), 155–163.Google Scholar
Miguel, C., ‘Menon’s identity in residually finite Dedekind domains’, J. Number Theory 137 (2014), 179–185.Google Scholar
Miguel, C., ‘A Menon-type identity in residually finite Dedekind domains’, J. Number Theory 164 (2016), 43–51.Google Scholar
Narkiewicz, W., Elementary and Analytic Theory of Algebraic Numbers (Polish Science Publishers, Warsaw, 1974).Google Scholar
Neumann, P., ‘A lemma that is not Burnside’s’, Math. Sci. 4 (1979), 133–141.Google Scholar
Sury, B., ‘Some number-theoretic identities from group actions’, Rend. Circ. Mat. Palermo 58 (2009), 99–108.Google Scholar
Tǎrnǎuceanu, M., ‘A generalization of Menon’s identity’, J. Number Theory 132 (2012), 2568–2573.Google Scholar
Zhang, X. and Ji, C. G., ‘Sums of generators of ideals in residue class ring’, J. Number Theory 174 (2017), 14–25.Google Scholar