Hostname: page-component-77c89778f8-gq7q9 Total loading time: 0 Render date: 2024-07-24T04:53:57.253Z Has data issue: false hasContentIssue false
  • English
  • Français

Algebraically closed commutative rings

Published online by Cambridge University Press:  12 March 2014

G. L. Cherlin
G. L. Cherlin*
Affiliation:
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Get access Rights & Permissions [Opens in a new window]

Extract

A commutative ring A is said to be algebraically closed if every finite system of polynomial equations and inequations in one or more variables with coefficients in A which has a solution in some (commutative) extension of A already has a solution in A. Abraham Robinson's study of model-theoretic forcing has provided powerful new tools for the study of algebraically closed structures in general, and will be applied here to the study of algebraically closed commutative rings. Familiarity with the model-theoretic notions connected with the study of algebraically closed structures is assumed; for background consult [1], [2], and [3].

Our main results are the following:

1. The theory of commutative rings with identity has no model companion in the sense of Robinson.

2. The Hilbert Nullstellensatz, suitably formulated for the class of algebraically closed commutative rings, holds for finitely generated polynomial ideals but fails for certain infinitely generated polynomial ideals.

3. If A is algebraically closed, then A/rad A need not be algebraically closed as a semiprime ring: If A is finitely generic then A/rad A is algebraically closed as a semiprime ring, but if A is infinitely generic then A/rad A is not algebraically closed as a semiprime ring.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 38 , Issue 3 , September 1973 , pp. 493 - 499
Copyright
Copyright © Association for Symbolic Logic 1973

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]Barwise, J. and Robinson, A., Completing theories by forcing, Annals of Mathematical Logic, vol. 2 (1970), pp. 119–142.CrossRefGoogle Scholar
[2]Cherlin, G., The model-companion of a class of structures, this Journal, vol. 37 (1972), pp. 546–556.Google Scholar
[3]Eklof, P. and Sabbagh, G., Model-completions and modules, Annals of Mathematical Logic, vol. 2 (1971), pp. 251–295. Appendix.CrossRefGoogle Scholar
[4]Lipschitz, L. and Saracino, D., The model companion of the theory of commutative rings without nilpotent elements, Proceedings of the American Mathematical Society, vol. 37 (1973).Google Scholar
[5]Macintyre, A., On algebraically closed division rings (to appear).Google Scholar
[6]Wheeler, W., Algebraically closed division rings, forcing, and the analytic hierarchy, Doctoral Dissertation, Yale University, New Haven, Conn., 1972.Google Scholar
[7]Hirschfeld, J., Existentially complete and generic structures in arithmetic, Doctoral Dissertation, Yale University, New Haven, Conn., 1972.Google Scholar