Let be a principal ideal domain and a torsion -module of finite type. We give an elementary proof of the fact that any -algebra automorphism of is inner.
Soit un anneau principal et un -module de torsion de type fini. Nous donnons une preuve élémentaire du fait que tout automorphisme de -algèbre de est intérieur.
@article{JTNB_2005__17_2_511_0, author = {Anne Cortella and Jean-Pierre Tignol}, title = {Le th\'eor\`eme de {Skolem-Noether} pour les modules sur des anneaux principaux}, journal = {Journal de Th\'eorie des Nombres de Bordeaux}, pages = {511--516}, publisher = {Universit\'e Bordeaux 1}, volume = {17}, number = {2}, year = {2005}, doi = {10.5802/jtnb.504}, zbl = {1092.13011}, mrnumber = {2211304}, language = {fr}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.504/} }
TY - JOUR TI - Le théorème de Skolem-Noether pour les modules sur des anneaux principaux JO - Journal de Théorie des Nombres de Bordeaux PY - 2005 DA - 2005/// SP - 511 EP - 516 VL - 17 IS - 2 PB - Université Bordeaux 1 UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.504/ UR - https://zbmath.org/?q=an%3A1092.13011 UR - https://www.ams.org/mathscinet-getitem?mr=2211304 UR - https://doi.org/10.5802/jtnb.504 DO - 10.5802/jtnb.504 LA - fr ID - JTNB_2005__17_2_511_0 ER -
Anne Cortella; Jean-Pierre Tignol. Le théorème de Skolem-Noether pour les modules sur des anneaux principaux. Journal de Théorie des Nombres de Bordeaux, Volume 17 (2005) no. 2, pp. 511-516. doi : 10.5802/jtnb.504. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.504/
[B1] R. Baer, Automorphism rings of primary abelian operator groups. Ann. Math. 44 (1943), 192–227. | MR: 8241 | Zbl: 0061.05405
[B2] R. Baer, Linear algebra and projective geometry. Academic Press (1952). | MR: 52795 | Zbl: 0049.38103
[F1] L. Fuchs, Infinite abelian groups, vol. 1. Academic Press (1970). | MR: 255673 | Zbl: 0209.05503
[F2] L. Fuchs, Infinite abelian groups, vol. 2. Academic Press (1973). | MR: 349869 | Zbl: 0257.20035
[G] R. Goebel, Endomorphism rings of abelian groups. Lecture notes in math. 1006 (1983), 340–353. | MR: 722628 | Zbl: 0516.20032
[I] I.M. Isaacs, Automorphisms of matrix algebras over commutative rings. Linear. Alg. Appli. 31 (1980), 215–231. | MR: 570392 | Zbl: 0434.16015
[K1] I. Kaplansky, Some results on abelian groups. Proc. Nat. Acad. Sci. USA 38 (1952), 538–540. | MR: 49189 | Zbl: 0047.25804
[K2] I. Kaplansky, Infinite abelian groups. Univ. Michigan Press (1954) ; rev. ed. 1969. | MR: 233887 | Zbl: 0194.04402
[Kn] M.-A. Knus, Algebres d’azumaya et modules projectifs. Commen. Math. Helv. 45 (1970), 372–383. | Zbl: 0205.34203
[L1] T.Y. Lam, A first course in non-commutative rings. Springer-Verlag (1991). | MR: 1125071 | Zbl: 0728.16001
[L2] 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
[L3] T.Y. Lam, Exercises in Classical Ring Theory. Springer-Verlag (1995). | MR: 1323431 | Zbl: 0823.16001
[M] A.V. Mikhalev, Isomorphisms and anti-isomorphisms of endomorphism rings of modules. Proc. first international Tainan-Moscow alg. workshop, Berlin (1996). | MR: 1443438 | Zbl: 0881.16017
[RZ] A. Rosenberg, D. Zelinsky, Automorphisms of separable algebras, Pacif. J. Math. 11 (1961), 1109–1117. | MR: 148709 | Zbl: 0116.02501
[Thz] J. Thevenaz, -algebras and modular representation theory. Oxford Sc. Publi. (1995). | MR: 1365077 | Zbl: 0837.20015
[W] K.G. Wolfson, Anti-isomorphisms of endomorphism rings of locally free modules. Math. Z. 202 (1989), 1951–1959. | MR: 1013081 | Zbl: 0655.16016
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