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Free division rings of fractions of crossed products of groups with Conradian left-orders

  • Joachim Gräter EMAIL logo
From the journal Forum Mathematicum

Abstract

Let D be a division ring of fractions of a crossed product F[G,η,α], where F is a skew field and G is a group with Conradian left-order . For D we introduce the notion of freeness with respect to and show that D is free in this sense if and only if D can canonically be embedded into the endomorphism ring of the right F-vector space F((G)) of all formal power series in G over F with respect to . From this we obtain that all division rings of fractions of F[G,η,α] which are free with respect to at least one Conradian left-order of G are isomorphic and that they are free with respect to any Conradian left-order of G. Moreover, F[G,η,α] possesses a division ring of fraction which is free in this sense if and only if the rational closure of F[G,η,α] in the endomorphism ring of the corresponding right F-vector space F((G)) is a skew field.


Communicated by Manfred Droste


References

[1] A. A. Bovdi, Crossed products of a semigroup and a ring, Dokl. Akad. Nauk SSSR 137 (1961), 1267–1269. Search in Google Scholar

[2] S. D. Brodskiĭ, Equations over groups, and groups with one defining relation, Sibirsk. Mat. Zh. 25 (1984), no. 2, 84–103. 10.1007/BF00971461Search in Google Scholar

[3] R. G. Burns and V. W. D. Hale, A note on group rings of certain torsion-free groups, Canad. Math. Bull. 15 (1972), 441–445. 10.4153/CMB-1972-080-3Search in Google Scholar

[4] A. Clay and D. Rolfsen, Ordered Groups and Topology, Grad. Stud. Math. 176, American Mathematical Society, Providence, 2016. 10.1090/gsm/176Search in Google Scholar

[5] P. M. Cohn, An Introduction to Ring Theory, Springer Undergrad. Math. Ser., Springer, London, 2000. 10.1007/978-1-4471-0475-9Search in Google Scholar

[6] P. Conrad, Right-ordered groups, Michigan Math. J. 6 (1959), 267–275. 10.1307/mmj/1028998233Search in Google Scholar

[7] W. Dicks, D. Herbera and J. Sánchez, On a theorem of Ian Hughes about division rings of fractions, Comm. Algebra 32 (2004), no. 3, 1127–1149. 10.1081/AGB-120027970Search in Google Scholar

[8] N. I. Dubrovin, Invertibility of the group ring of a right-ordered group over a division ring (in Russian), Mat. Zametki 42 (1987), no. 4, 508–518, 622; translation in Math. Notes 42 (1982), no. 3–4, 781–786. Search in Google Scholar

[9] N. I. Dubrovin, Rational closures of group rings of left-ordered groups (in Russian), Mat. Sb. 184 (1993), no. 7, 3–48; translation in Russian Acad. Sci. Sb. Math. 79 (2) (1993), 231–263. 10.1070/SM1994v079n02ABEH003498Search in Google Scholar

[10] N. I. Dubrovin, Rational closures of group rings of left-ordered groups, SM-DU-254, Duisburg, 1994. 10.1070/SM1994v079n02ABEH003498Search in Google Scholar

[11] N. I. Dubrovin, Formal sums and power series over a group (in Russian), Mat. Sb. 191 (2000), no. 7, 13–30; translation in Sb. Math. 191 (2000), no. 7–8, 955-971. 10.1070/SM2000v191n07ABEH000490Search in Google Scholar

[12] N. I. Dubrovin, J. Gräter and T. Hanke, Complexity of elements in rings, Algebr. Represent. Theory 6 (2003), no. 1, 33–45. 10.1023/A:1022320103094Search in Google Scholar

[13] J. Gräter, Left-orderings, power series, valuations, unpublished (2015), Universität Potsdam. Search in Google Scholar

[14] J. Gräter and R. P. Sperner, On embedding left-ordered groups into division rings, Forum Math. 27 (2015), no. 1, 485–518. 10.1515/forum-2012-0070Search in Google Scholar

[15] G. Higman, The units of group-rings, Proc. Lond. Math. Soc. (2) 46 (1940), 231–248. 10.1112/plms/s2-46.1.231Search in Google Scholar

[16] O. Hölder, Die Axiome der Quantität und die Lehre vom Maß, Ber. Verh. Kgl. Sächs. Ges. Wiss. Leipzig Math.-Phys. Kl. 53 (1901), 1–64. Search in Google Scholar

[17] I. Hughes, Division rings of fractions for group rings, Comm. Pure Appl. Math. 23 (1970), 181–188. 10.1002/cpa.3160230205Search in Google Scholar

[18] I. Hughes, Division rings of fractions for group rings. II, Comm. Pure Appl. Math. 25 (1972), 127–131. 10.1002/cpa.3160250202Search in Google Scholar

[19] A. Jaikin-Zapirain and D. López-Álvarez, The strong Atiyah conjecture for one-relator groups, preprint (2018), https://arxiv.org/abs/1810.12135. Search in Google Scholar

[20] F. Jakobs, Dubrovin-rings and their connection to Hughes-free skew fields of fractions, Doctoral Dissertation, Universität Potsdam, 2019. Search in Google Scholar

[21] V. M. Kopytov and N. Y. Medvedev, Right-ordered groups, Consultants Bureau, New York, 1996. Search in Google Scholar

[22] P. A. Linnell, Noncommutative localization in group rings, Non-commutative Localization in Algebra and Topology, London Math. Soc. Lecture Note Ser. 330, Cambridge University, Cambridge (2006), 40–59. 10.1017/CBO9780511526381.010Search in Google Scholar

[23] A. Navas, On the dynamics of (left) orderable groups, Ann. Inst. Fourier (Grenoble) 60 (2010), no. 5, 1685–1740. 10.5802/aif.2570Search in Google Scholar

[24] B. H. Neumann, On ordered division rings, Trans. Amer. Math. Soc. 66 (1949), 202–252. 10.1090/S0002-9947-1949-0032593-5Search in Google Scholar

[25] D. S. Passman, Infinite Crossed Products, Pure Appl. Math. 135, Academic Press, Boston, 1989. Search in Google Scholar

[26] A. Rhemtulla and D. Rolfsen, Local indicability in ordered groups: braids and elementary amenable groups, Proc. Amer. Math. Soc. 130 (2002), no. 9, 2569–2577. 10.1090/S0002-9939-02-06413-4Search in Google Scholar

[27] M. Simon, Stetige und v-verträgliche Automorphismen, Diplomarbeit, Universität Potsdam, 2009. Search in Google Scholar

Received: 2019-09-24
Published Online: 2020-02-11
Published in Print: 2020-05-01

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