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IDEAL CHAINS IN RESIDUALLY FINITE DEDEKIND DOMAINS

  • YU-JIE WANG (a1), YI-JING HU (a2) and CHUN-GANG JI (a3)
Abstract

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.

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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).

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[1] 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), 4250.
[2] Li, Y. and Kim, D., ‘Menon-type identities derived from actions of subgroups of general linear groups’, J. Number Theory 179 (2017), 97112.
[3] Menon, P. K., ‘On the sum ∑(a - 1, n)[(a, n) = 1]’, J. Indian Math. Soc. 29 (1965), 155163.
[4] Miguel, C., ‘Menon’s identity in residually finite Dedekind domains’, J. Number Theory 137 (2014), 179185.
[5] Miguel, C., ‘A Menon-type identity in residually finite Dedekind domains’, J. Number Theory 164 (2016), 4351.
[6] Narkiewicz, W., Elementary and Analytic Theory of Algebraic Numbers (Polish Science Publishers, Warsaw, 1974).
[7] Neumann, P., ‘A lemma that is not Burnside’s’, Math. Sci. 4 (1979), 133141.
[8] Sury, B., ‘Some number-theoretic identities from group actions’, Rend. Circ. Mat. Palermo 58 (2009), 99108.
[9] Tǎrnǎuceanu, M., ‘A generalization of Menon’s identity’, J. Number Theory 132 (2012), 25682573.
[10] Zhang, X. and Ji, C. G., ‘Sums of generators of ideals in residue class ring’, J. Number Theory 174 (2017), 1425.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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