Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-02-04T07:30:26.717Z Has data issue: false hasContentIssue false
  • English
  • Français

The ℵ1-categoricity of strictly upper triangular matrix rings over algebraically closed fields

Published online by Cambridge University Press:  12 March 2014

Bruce I. Rose
Bruce I. Rose*
Affiliation:
University of Notre Dame, Notre Dame, Indiana 46556
Get access Rights & Permissions [Opens in a new window]

Abstract

Let n ≥ 3. The following theorems are proved.

Theorem. The theory of the class of strictly upper triangular n × n matrix rings over fields is finitely axiomatizable.

Theorem. If R is a strictly upper triangular n × n matrix ring over a field K, then there is a recursive map σ from sentences in the language of rings with constants for K into sentences in the language of rings with constants for R such that K ⊨ φ if and only if R φ σ(φ).

Theorem. The theory of a strictly upper triangular n × n matrix ring over an algebraically closed field is ℵ1-categorical.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 43 , Issue 2 , June 1978 , pp. 250 - 259
Copyright
Copyright © Association for Symbolic Logic 1978

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

[1]Chang, C. C. and Keisler, H. J., Model theory, North-Holland, Amsterdam, 1972.Google Scholar
[2]Cherlin, G. L. and Reineke, J., Categoricity and stability of commutative rings, Annals of Mathematical Logic, vol. 9 (1976),pp. 367–400.CrossRefGoogle Scholar
[3]Felgner, U., Ringe, deren theorien ℵ1-categorich sind, Fundamenta Mathematicae, vol. 72 (1975), pp. 331–346.CrossRefGoogle Scholar
[4]Macintyre, A., On ω1-categorical theories of Abelian groups, Fundamenta Mathematicae, vol. 70 (1970), pp. 253–270.CrossRefGoogle Scholar
[5]Macintyre, A., On ω1-categorical theories of fields, Fundamenta Mathematicae, vol. 71 (1971), pp. 1–25.CrossRefGoogle Scholar
[6]Mal'cev, A. I., A correspondence between rings and groups, A. I. Mal'cev, The metamathematics of algebraic systems, North-Holland, Amsterdam, 1971, pp. 124–137.Google Scholar
[7]Sacks, G. E., Saturated model theory, W. A. Benjamin, Reading, Massachusetts, 1972.Google Scholar
[8]Shelah, S., Every two elementarily equivalent models have isomorphic ultrapowers, Israel Journal of Mathematics, vol. 10 (1971), pp. 224–233.CrossRefGoogle Scholar
[9]Zil'ber, B. I., Rings with ℵ1-categorical theories, Algebra and Logic, vol. 13 (1974), pp. 95–104.CrossRefGoogle Scholar