Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-16T20:53:43.473Z Has data issue: false hasContentIssue false
  • English
  • Français

ω-STABILITY AND MORLEY RANK OF BILINEAR MAPS, RINGS AND NILPOTENT GROUPS

Published online by Cambridge University Press:  19 June 2017

ALEXEI G. MYASNIKOV  and
MAHMOOD SOHRABI
ALEXEI G. MYASNIKOV
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES STEVENS INSTITUTE OF TECHNOLOGY HOBOKEN, NJ 07030, USAE-mail: amiasnikov@gmail.com
MAHMOOD SOHRABI
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES STEVENS INSTITUTE OF TECHNOLOGY HOBOKEN, NJ 07030, USAE-mail: msohrab1@stevens.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we study the algebraic structure of ω-stable bilinear maps, arbitrary rings, and nilpotent groups. We will also provide rather complete structure theorems for the above structures in the finite Morley rank case.

Keywords

bilinear maplargest ring of scalarsringnilpotent groupω-stabilityMorley rank
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 2 , June 2017 , pp. 754 - 777
Copyright
Copyright © The Association for Symbolic Logic 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Sohrabi, M. and Myasnikov, A. G., Elementary coordinatization of finitely generated nilpotent groups , preprint, arXiv:1311.1391v3.Google Scholar
Sohrabi, M. and Myasnikov, A. G., Groups elementarily equivalent to a free nilpotent group of finite rank . Annals of Pure and Applied Logic. vol. 162 (2011), no. 11, pp. 916–933.Google Scholar
Nesin, A. and Borovik, A. V., Groups of Finite Morley Rank, Oxford Logic Guides, vol. 26, Oxford University Press, New York, 1994.Google Scholar
Baudisch, A., A new uncountably categorical group. Transactions of the American Mathematical Society. vol. 348 (1998), no. 10, pp. 3889–3940.Google Scholar
Baumslag, G., Lecture Notes on Nilpotent Groups , Regional Conference Series in Mathematics, no. 2, American Mathematical Society, Providence, RI, 1971.Google Scholar
Frécon, O., Around unipotence in groups of finite morley rank . Journal of Group Theory, vol. 9 (2006), no. 3, pp. 341–359.CrossRefGoogle Scholar
Frécon, O., Conjugacy of carter subgroups in groups of finite morley rank . Journal of Mathematical Logic, vol. 8 (2008), pp. 41–92.Google Scholar
Reineke, J. and Cherlin, G. L., Categoricity and stability of commutative rings . Annals of Mathematical Logic, vol. 10 (1976), pp. 367–399.Google Scholar
Hall, P., Nilpotent Groups, Notes of lectures given at the Canadian Mathematical Congress, University of Alberta, 1957.Google Scholar
Macintyre, A., On ω 1 -categorical theories of abelian groups . Fundamenta Mathematicae, vol. 70 (1970), pp. 253–270.Google Scholar
Macintyre, A., On ω 1 -categorical theories of fields. Fundamenta Mathematicae , vol. 71 (1971), pp. 1–25.Google Scholar
Myasnikov, A. G., Elementary theory of a module over a local ring . Siberian Mathematical Journal, vol. 30 (1989), no. 3, pp. 72–83.Google Scholar
Myasnikov, A. G., Definable invariants of bilinear mappings . Siberian Mathematical Journal, vol. 31 (1990), no. 1, pp. 89–99.Google Scholar
Myasnikov, A. G., The structure of models and a criterion for the decidability of complete theories of finite-dimensional algebras . Mathematics of USSR-Izvestia, vol. 34 (1990), no. 2, pp. 389–407.Google Scholar
Myasnikov, A. G., The theory of models of bilinear mappings . Siberian Mathematical Journal, vol. 31 (1990), no. 3, pp. 439–451.CrossRefGoogle Scholar
Nesin, A., Poly-separated and ω-stable nilpotent groups , this Journal, vol. 56 (1991), pp. 694–699.Google Scholar
Quillen, D., Rational homotopy theory . Annals of Mathematics, vol. 90 (1960), pp. 205–295.Google Scholar
Sohrabi, M., On the elementary theories of free nilpotent lie algebras and free nilpotent groups, Ph.D. thesis, Carleton University, Ottawa/Canada, 2009.Google Scholar
Stewart, I., An algebraic treatment of mal’cevs theorem concerning nilpotent lie groups and their lie algebras . Compositio Mathematica, vol. 22 (1970), pp. 289–312.Google Scholar
Wilson, J. S. and Altinel, T., On the linearity of torsion-free nilpotent groups of finite morley rank . Proceedings of the American Mathematical Society, 2009.Google Scholar
Warfield, R. B., Nilpotent Groups, Lecture Notes in Mathematics, vol. 513, Springer-Verlag Berlin, 1976.Google Scholar
Zariski, O. and Samuel, P., Commutative Algebra. II, Van Nostrand, Princeton, 1960.Google Scholar
Zilber, B. I., Rings with-categorical theories . Algebra and Logic, vol. 13 (1974), pp. 95–104.Google Scholar
Zilber, B. I., Uncountably categorical nilpotent groups and lie algebras . Soviet Mathematics (Iz. VUZ), vol. 26 (1982), no. 5, pp. 98–99.Google Scholar