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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
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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

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References

REFERENCES

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[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