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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
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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

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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