Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-26T12:32:04.671Z Has data issue: false hasContentIssue false
  • English
  • Français

Homomorphisms of two-dimensional linear groups over Laurent polynomial rings and Gaussian domains

Published online by Cambridge University Press:  24 October 2008

Yu Chen
Yu Chen
Affiliation:
Institute of Systems Science, Academia Sinica, Beijing, China Mathematics Institute, University of Warwick, Coventry CV4 7AL
Get access Rights & Permissions [Opens in a new window]

Extract

Let GL2(R) be the general linear group of 2 × 2 invertible matrices in M2(R) over a commutative ring R with 1 and SL2(R) be the special linear group consisting of 2 × 2 matrices over R with determinant 1. In this paper we determine the homomorphisms from GL2 and SL2, as well as their projective groups, over Laurent polynomial rings to those groups over Gaussian domains, i.e. unique factorization domains (cf. Theorems 1, 2, 3 below). We also consider more generally the homomorphisms of non-projective groups over commutative rings containing a field which are generated by their units (cf. Theorems 4 and 5). So far the homomorphisms of two-dimensional linear groups over commutative rings have only been studied in some specific cases. Landin and Reiner[7] obtained the automorphisms of GL2(R), where R is a Euclidean domain generated by its units. When R is a type of generalized Euclidean domain with a degree function and with units of R and 0 forming a field, Cohn[3] described the automorphisms of GL2(R). Later, Cohn[4] applied his methods to the case of certain rings of quadratic integers. Dull[6] has considered the automorphisms of GL2(R) and SL2(R), along with their projective groups, provided that R is a GE-ring and 2 is a unit in R. McDonald [9] examined the automorphisms of GL2(R) when R has a large unit group. The most recent work of which we are aware is that of Li and Ren[8] where the automorphisms of E2(R) and GE2(R) were determined for any commutative ring R in which 2, 3 and 5 are units.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 109 , Issue 2 , March 1991 , pp. 287 - 297
Copyright
Copyright © Cambridge Philosophical Society 1991

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]Bass, H.. Algebraic K-theory (Benjamin, 1968).Google Scholar
[2]Chen, Y.. Homomorphisms of two dimensional linear groups. Comm. Algebra. 18 (1990), 2383–2396.CrossRefGoogle Scholar
[3]Cohn, P. M.. On the structure of GL2 of a ring. Inst. Hautes Études Sci. Publ. Math. 30 (1966), 5–53.CrossRefGoogle Scholar
[4]Cohn, P. M.. Automorphisms of two dimensional linear groups over Euclidean domains. J. London Math. Soc. (2) 1 (1969), 279–292.CrossRefGoogle Scholar
[5]Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A.. Atlas of Finite Groups (Clarendon Press, 1985).Google Scholar
[6]Dull, M. H.. Automorphisms of two dimensional linear groups over integral domains. Amer. J. Math. 96 (1974), 1–40.CrossRefGoogle Scholar
[7]Landin, J. and Reiner, I.. Automorphisms of two dimensional general linear groups over a Euclidean ring. Proc. Amer. Math. Soc. 9 (1958), 209–216.Google Scholar
[8]Li, F. and Ren, H.. The automorphisms of two dimensional linear groups over commutative rings. Chinese Ann. Math. Ser. B 10 (1989), 50–57.Google Scholar
[9]McDonald, B.. Aut(GL2(R)) for rings with many units. Comm. Algebra 9 (1981), 205–220.CrossRefGoogle Scholar