On reduced rings and number theory
Abstract
In this note we exhibit a connection between theory of associative rings and number theory by an example of necessary and sufficient conditions under which the integral group ring of a finite group is reduced.
Keywords
reduced ring; level of a field; prime numbers
References
2. A. Brauer, A note on a number theoretical paper of Sierpiński, Proc. Amer. Math. Soc., 11 (1960), 406-409.
3. P. Chowla, On the representation of -1 as sum of squares in a cyclotomic field, J. Number Theory, 1 (1969), 208-210.
4. P. Chowla, S. Chowla, Determination of the stufe of certain cyclotomic fields, J. Number Theory, 2 (1970), 271-272.
5. J. Krempa, On finite generation of unit group for group rings, Groups '93 Galway/St Andrews, vol. 2, London Math. Soc. Lecture Note 212, Cambridge University Press, Cambridge (1995), 352-367.
6. J. Krempa, Rings with periodic unit groups, Abelian groups and modules, A. Facchini, C. Menini, Kluwer Academic Publishers, Dordrecht (1995), 313-321.
7. J. Krempa, Some examples of reduced rings, Algebra Colloqium, 3 (1996), 289-300.
8. R. Kučera, K. Szymiczek, Witt equivalence of cyclotomic fields, Math. Slovaca, 42 (1992), 663-676.
9. T.Y. Lam, The algebraic theory of quadratic forms, W.A. Benjamin Inc., Reading Massachusetts (1973).
10. Z.S. Marciniak, S.K. Sehgal, Units in group rings and geometry, Methods in Ring Theory, V. Drensky, A. Giambrouno, S.K. Sehgal, Marcel Dekker Inc., New York (1998), 185-198.
11. C. Moser, Representation de -1 comme somme de carres dans un corps cyclotomique quelconque, J. Number Theory, 5 (1973), 139-141.
12. W. Narkiewicz, Elementary and analytic theory of algebraic numbers, Second ed., PWN and Springer-Verlag, Warszawa-Berlin-Heidelberg-New York (1990).
13. D.J.S. Robinson, A course in the theory of groups, 2nd, extended edition, Springer-Verlag, Berlin (1996).
14. L.H. Rowen, Ring theory, vol. I, Academic Press, New York (1988).
15. S.K. Sehgal, Topics in group rings, Marcel Dekker Inc., New York (1978).
16. W. Sierpiński, Sur une décomposition des nombres premiers en deux classes, Colloq. Math., 10 (1958), 81-83.
17. W. Sierpiński, Elementary theory of numbers, 2nd edition, revised by A. Schinzel, PWN, Warszawa (1987).
18. K. Szymiczek, Bilinear algebra. An introduction to the algebraic theory of quadratic forms, Algebra, Logic and Applications Series Volume 7, Gordon and Breach Science Publishers, Amsterdam (1997).
Instytut Matematyki, Uniwersytet Warszawski Poland
The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.
- License
This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license. - Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. - User Rights
Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor. - Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.
