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Automorphisms of $ {\rm GL}\sb{n}(R)$


Author: Bernard R. McDonald
Journal: Trans. Amer. Math. Soc. 215 (1976), 145-159
MSC: Primary 20G35
DOI: https://doi.org/10.1090/S0002-9947-1976-0382467-1
MathSciNet review: 0382467
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Abstract: Let R be a commutative ring and S a multiplicatively closed subset of R having no zero divisors. The pair $ \langle R,S\rangle $ is said to be stable if the ring of fractions of R, $ {S^{ - 1}}R$, defined by S is a ring for which all finitely generated projective modules are free. For a stable pair $ \langle R,S\rangle $ assume 2 is a unit in R and V is a free R-module of dimension $ \geqslant 3$. This paper examines the action of a group automorphism of $ GL(V)$ (the general linear group) on the elementary matrices relative to a basis B of V. In the case that R is a local ring, a Euclidean domain, a connected semilocal ring or a Dedekind domain whose quotient field is a finite extension of the rationals, we obtain a description of the action of the automorphism on all elements of $ GL(V)$.


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  • [1] H. Bass, Algebraic K-theory, Conf. Board Math. Sci., No. 20 Amer. Math. Soc., Providence, R. I., 1974.
  • [2] -, Algebraic K-theory, Benjamin, New York, 1968. MR 40 #2736. MR 0249491 (40:2736)
  • [3] N. Bourbaki, Commutative algebra, Addison-Wesley, Reading, Mass., (1972). (Éléments de Mathématique. Algèbre Commutative, 1964, 1965, 1968, 1969, Hermann, Paris). MR 0360549 (50:12997)
  • [4] P. M. Cohn, On the structure of $ {\text{GL}_2}$ of a ring, Inst. Hautes Études Sci. Publ. Math. No. 30 (1966), 5-53. MR 34 #7670. MR 0207856 (34:7670)
  • [5] G. Cooke, A weakening of the Euclidean property for integral domains and applications to algebraic number theory (to appear).
  • [6] J. Dieudonne, On the automorphisms of the classical groups, Mem. Amer. Math. Soc. No. 2 (1951). MR 13, 531. MR 0045125 (13:531e)
  • [7] L. K. Hua and I. Reiner, Automorphisms of the unimodular group, Trans. Amer. Math. Soc. 71 (1951), 331-348. MR 13, 328. MR 0043847 (13:328f)
  • [8] E. M. Keenan, On the automorphisms of classical groups over local rings, Thesis, M. I. T., 1965.
  • [9] J. Landin and I. Reiner, Automorphisms of the general linear group over a principal ideal domain, Ann. of Math. (2) 65 (1957), 519-526. MR 19, 388. MR 0087666 (19:388f)
  • [10] J. Milnor (with D. Husemoller), Symmetric bilinear forms, Princeton, N. J., 1972. MR 0506372 (58:22129)
  • [11] O. T. O'Meara, The automorphisms of the linear groups over any integral domain, J. Reine Angew. Math. 223 (1966), 56-100. MR 33 #7427. MR 0199278 (33:7427)
  • [12] -, Lectures on linear groups, Conf. Board Math. Sci. No. 22, Amer. Math. Soc., Providence, R. I., 1974. MR 0349862 (50:2355)
  • [13] M. Ojanguren and R. Sridharan, A note on the fundamental theorem of projective geometry, Comment. Math. Helv. 44 (1969), 310-315. MR 39 #4133. MR 0242806 (39:4133)
  • [14] J. Pomfret and B. R. McDonald, Automorphisms of $ {\text{GL}_n}(R)$, R a local ring, Trans. Amer. Math. Soc. 173 (1972), 379-388. MR 46 #9190. MR 0310087 (46:9190)
  • [15] O. Schreier and B. L. van der Waerden, Die Automorphismen der projektiven Gruppen, Abh. Math. Sem. Univ. Hamburg 6 (1928), 303-332.
  • [16] Shih-chien Yen, Linear groups over a ring, Acta Math. Sinica 15 (1965), 455-468 = Chinese Math.-Acta 7 (1965), 163-179. MR 36 #5237. MR 0222185 (36:5237)
  • [17] M. Dull, Automorphisms of the two-dimensional linear groups over domains, Amer. J. Math. 96 (1974), 1-40. MR 0357638 (50:10106)

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DOI: https://doi.org/10.1090/S0002-9947-1976-0382467-1
Keywords: Automorphisms, general linear group, commutative ring
Article copyright: © Copyright 1976 American Mathematical Society

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