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The Skolem-Noether theorem for modules over principal rings. (Le théorème de Skolem-Noether pour les modules sur les anneaux principaux.) (French) Zbl 1092.13011

Summary: Let \(k\) be a principal ideal domain and \(M\) a torsion \(k\)-module of finite type. We give an elementary proof of the fact that any \(k\)-algebra automorphism of \(R=\text{End}_kM\) is inner.

MSC:

16W20 Automorphisms and endomorphisms
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References:

[1] R. Baer, Automorphism rings of primary abelian operator groups. Ann. Math. 44 (1943), 192-227. · Zbl 0061.05405
[2] R. Baer, Linear algebra and projective geometry. Academic Press (1952). · Zbl 0049.38103
[3] L. Fuchs, Infinite abelian groups, vol. 1. Academic Press (1970). · Zbl 0209.05503
[4] L. Fuchs, Infinite abelian groups, vol. 2. Academic Press (1973). · Zbl 0257.20035
[5] R. Goebel, Endomorphism rings of abelian groups. Lecture notes in math. 1006 (1983), 340-353. · Zbl 0516.20032
[6] I.M. Isaacs, Automorphisms of matrix algebras over commutative rings. Linear. Alg. Appli. 31 (1980), 215-231. · Zbl 0434.16015
[7] I. Kaplansky, Some results on abelian groups. Proc. Nat. Acad. Sci. USA 38 (1952), 538-540. · Zbl 0047.25804
[8] I. Kaplansky, Infinite abelian groups. Univ. Michigan Press (1954) ; rev. ed. 1969. · Zbl 0194.04402
[9] M.-A. Knus, Algebres d’azumaya et modules projectifs. Commen. Math. Helv. 45 (1970), 372-383. · Zbl 0205.34203
[10] T.Y. Lam, A first course in non-commutative rings. Springer-Verlag (1991). · Zbl 0728.16001
[11] T.Y. Lam, Modules with isomorphic multiples and rings with isomorphic matrix rings, a survey. Monographies de l’enseign. math., Genève 35 (1999). · Zbl 0962.16002
[12] T.Y. Lam, Exercises in Classical Ring Theory. Springer-Verlag (1995). · Zbl 0823.16001
[13] A.V. Mikhalev, Isomorphisms and anti-isomorphisms of endomorphism rings of modules. Proc. first international Tainan-Moscow alg. workshop, Berlin (1996). · Zbl 0881.16017
[14] A. Rosenberg, D. Zelinsky, Automorphisms of separable algebras, Pacif. J. Math. 11 (1961), 1109-1117. · Zbl 0116.02501
[15] J. Thevenaz, \(G\)-algebras and modular representation theory. Oxford Sc. Publi. (1995). · Zbl 0837.20015
[16] K.G. Wolfson, Anti-isomorphisms of endomorphism rings of locally free modules. Math. Z. 202 (1989), 1951-1959. · Zbl 0655.16016
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