Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T08:27:04.335Z Has data issue: false hasContentIssue false
  • English
  • Français

Algebraically Closed Regular Rings

Published online by Cambridge University Press:  20 November 2018

Andrew B. Carson
Andrew B. Carson*
Affiliation:
University of Saskatchewan, Saskatoon, Saskatchewan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper all rings are commutative and have a unity. All ring homomorphisms preserve the unity. We let L denote the standard language for rings with two distinct constants, 0 and 1, playing the role of the zero and the unity respectively. A ring is regular if it satisfies the axiom (∀r) (∃r′)(rr′r = r) and it is algebraically closed if, for each integer n ≧ 1, it satisfies the sentence

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 5 , October 1974 , pp. 1036 - 1049
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Arens, R. and Kaplansky, I., Topological representations of algebras, Trans. Amer. Math. Soc. 63 (1948), 457–481.Google Scholar
2. Bell, J. L. and Slomson, A. B., Models and ultraproducts: an introduction (North-Holland, Amsterdam, 1969).Google Scholar
3. Bredon, G. E., Sheaf theory (McGraw-Hill, New York, 1967).Google Scholar
4. Carson, A. B., Representation of semi-simple algebraic algebras, J. Algebra 24 (1973), 245–257.Google Scholar
5. Carson, A. B., The model completion of the theory of commutative regular rings, J. Algebra 27 (1973), 136–146.Google Scholar
6. Carson, A. B., Partially self-infective regular rings, Can. Math. Bull, (to appear).Google Scholar
7. Comer, S. D., Elementary properties of structures of sections, Vanderbilt University, Nashville (preprint).Google Scholar
8. Ersov, Y., The decidability of the elementary theory of Boolean algebras, Algebra and Logic 3 (1964), 17–38.Google Scholar
9. Feferman, S. and Vaught, R. L., The first order properties of products of algebraic systems, Fund. Math. 47 (1959), 57–103.Google Scholar
10. Lambek, J., Lectures on rings and modules (Blaisdell, Waltham, Massachusetts, 1966).Google Scholar
11. Pierce, R. S., Modules over commutative regular rings, Mem. Amer. Math. Soc. 70 (1966).Google Scholar
12. Robinson, A., Introduction to model theory and to the metamathematics of algebra (North- Holland, Amsterdam, 1963).Google Scholar
13. Robinson, A. and E. Zakon, Elementary properties of ordered Abelian groups, Trans. Amer. Math. Soc. 96 (1960), 222–236.Google Scholar