Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-04T21:02:40.280Z Has data issue: false hasContentIssue false
  • English
  • Français

Artin's problem for skew field extensions

Published online by Cambridge University Press:  24 October 2008

A. H. Schofield
A. H. Schofield
Affiliation:
Trinity College, Cambridge
Get access Rights & Permissions [Opens in a new window]

Extract

For a commutative field extension, L ⊃ K, it is clear that a left basis of L over K; is also a right basis of L over K; however, for an extension of skew fields, this may easily fail, though it is hard to determine whether the right and left dimension may be different. Cohn ([4], ch. 5), however, was able to find extensions of skew fields such that the left and right dimensions were an arbitrary pair of cardinals subject only to the restrictions that neither were 1 and at least one of them was infinite. In this paper, I shall present a new approach that allows us to construct extensions of skew fields such that the left and right dimensions are arbitrary integers not equal to 1. In a subsequent paper, [7], I shall present related results and consequences; in particular, there is a construction of a hereditary artinian ring of finite representation type corresponding to the Coxeter diagram I2(5) answering the question raised by Dowbor, Ringel and Simson[5].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 97 , Issue 1 , January 1985 , pp. 1 - 6
Copyright
Copyright © Cambridge Philosophical Society 1985

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]Bergman, G. M.. Modules over coproducts of rings. Trans. Amer. Math. Soc. 200 (1974), 1–32.CrossRefGoogle Scholar
[2]Bergman, G. M.. Coproducts and some universal ring constructions. Trans. Amer. Math. Soc. 200 (1974), 33–88.CrossRefGoogle Scholar
[3]Cohn, P. M.. Free rings and their relations, L.M.S. Monographs no. 2 (Academic Press, 1971).Google Scholar
[4]Cohn, P. M.. Skew field constructions, L.M.S. Lecture Notes no. 27 (Cambridge University Press, 1977).Google Scholar
[5]Dowbor, P., Ringel, C. M. and Simson, D.. Hereditary artinian rings of finite representation type. In Representation Theory, II, Lecture Notes in Math. vol. 832 (Springer-Verlag, 1980), 232–241.Google Scholar
[6]Schofield, A. H.. Representations of rings over skew fields. (To appear in the L.M.S. Lecture Note Series.)Google Scholar
[7]Schofield, A. H.. Simple artinian extensions and hereditary artinian rings. (To appear.)Google Scholar