Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-12T20:32:21.572Z Has data issue: false hasContentIssue false
  • English
  • Français

Presentations of skew fields. I. Existentially closed skew fields and the Nullstellensatz

Published online by Cambridge University Press:  24 October 2008

P. M. Cohn
P. M. Cohn
Affiliation:
Bedford College, Regents Park, London NW1 4NS
Get access Rights & Permissions [Opens in a new window]

Extract

1. Introduction. The Nullstellensatz in commutative algebraic geometry may be described as a means of studying certain commutative rings (viz. affine algebras) by their homomorphisms into algebraically closed fields, and a number of attempts have been made to extend the result to the non-commutative case. In particular, Amitsur and Procesi have studied the case of general rings, with homomorphisms into matrix rings over commutative fields ((1), (2)) and Procesi has obtained more precise results for homomorphisms of PI-rings (11). Since a finite-dimensional division algebra can always be embedded in a matrix ring over a field, this includes the case of skew fields that are finite-dimensional over their centre, but it tells us nothing about general skew fields.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 77 , Issue 1 , January 1975 , pp. 7 - 19
Copyright
Copyright © Cambridge Philosophical Society 1975

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)Amitsur, S. A.A generalization of Hilbert's Nullstellensatz. Proc. Amer. Math. Soc. 8 (1957), 649–656.Google Scholar
(2)Amitsur, S. A. and Procesi, C.Jacobson rings and Hilbert algebras with polynomial identities. Ann. Mat. Pura Appl. (IV), 71 (1966), 61–72.CrossRefGoogle Scholar
(3)Boffa, M. and Praag, P. v.Sur les corps génériques, C.R. Acad. Sci. Paris Sér. A, 274 (1972), 1325–1327.Google Scholar
(4)Cohn, P. M.Rings with a weak algorithm, Trans. Amer. Math. Soc. 109 (1963), 332–356.CrossRefGoogle Scholar
(5)Cohn, P. M.The embedding of firs in skew fields. Proc. Lond. Math. Soc. (3), 23 (1971), 193–213.CrossRefGoogle Scholar
(6)Cohn, P. M.Free rings and their relations, LMS monographs No. 2 (Academic Press, London and New York, 1971).Google Scholar
(7)Cohn, P. M.Free products of skew fields, J. Austral. Math. Soc. 16 (1973), 300–308.CrossRefGoogle Scholar
(8)Kueker, D. W. Definability, automorphisms and infinitary languages, in Syntax and semantics in infinitary languages (ed. Barwise, J), Springer Lecture Notes No. 72 (Springer, Berlin 1968).Google Scholar
(9)Macintyre, A.On algebraically closed division rings, to appear in Annals of Math. Logic.Google Scholar
(10)Nuemann, B. H. The isomorphism problem for algebraically closed groups, in ‘Word problems’ (ed. Boone, W. W. et al. ) (North Holland, Amsterdam, 1973).Google Scholar
(11)Procesi, C.A non-commutative Hilbert Nullstellensatz. Rend. Mat. e Appl. 25 (1966), 17–21.Google Scholar
(12)Robinson, A.On the notion of algebraic closedness for noncommutative groups and fields. J. Symbolic Logic 36 (1971), 441–444.CrossRefGoogle Scholar
(13)Wheeler, W. H. Algebraically closed division rings, forcing and the analytical hierarchy, Ph.D. thesis (Yale University 1972).Google Scholar