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Skolem-Noether algebras. (English) Zbl 1439.16016
Summary: An algebra $$S$$ is called a Skolem-Noether algebra (SN algebra for short) if for every central simple algebra $$R$$, every homomorphism $$R \rightarrow R \otimes S$$ extends to an inner automorphism of $$R \otimes S$$. One of the important properties of such an algebra is that each automorphism of a matrix algebra over $$S$$ is the composition of an inner automorphism with an automorphism of $$S$$. The bulk of the paper is devoted to finding properties and examples of SN algebras. The classical Skolem-Noether theorem implies that every central simple algebra is SN. In this article it is shown that actually so is every semilocal, and hence every finite-dimensional algebra. Not every domain is SN, but, for instance, unique factorization domains, polynomial algebras and free algebras are. Further, an algebra $$S$$ is SN if and only if the power series algebra $$S [[\xi]]$$ is SN.

##### MSC:
 16K20 Finite-dimensional division rings 16W20 Automorphisms and endomorphisms 16L30 Noncommutative local and semilocal rings, perfect rings 16U10 Integral domains (associative rings and algebras)
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##### References:
 [1] Benkart, G. M.; Osborn, J. M., Derivations and automorphisms of nonassociative matrix algebras, Trans. Amer. Math. Soc., 263, 411-430, (1981) · Zbl 0453.16020 [2] Brešar, M., Introduction to noncommutative algebra, Universitext, (2014), Springer · Zbl 1334.16001 [3] Brešar, M., Derivations of tensor products of nonassociative algebras, Linear Algebra Appl., 530, 244-252, (2017) · Zbl 1368.17002 [4] Bunce, J. W., Automorphisms and tensor products of algebras, Proc. Amer. Math. Soc., 44, 93-95, (1974) · Zbl 0257.46090 [5] Cohn, P. M., Free ideal rings and localization in general rings, New Mathematical Monographs, vol. 3, (2006), Cambridge University Press Cambridge · Zbl 1114.16001 [6] Effros, E. G.; Rosenberg, J., $$C^\ast$$-algebras with approximately inner flip, Pacific J. Math., 77, 417-443, (1978) · Zbl 0412.46052 [7] Gille, P.; Szamuely, T., Central simple algebras and Galois cohomology, Cambridge Studies in Advanced Mathematics, vol. 101, (2006), Cambridge University Press · Zbl 1137.12001 [8] Herstein, I. N., Noncommutative rings, Carus Math. Monogr., vol. 15, (1968), The Mathematical Association of America · Zbl 0177.05801 [9] Isaacs, I. M., Automorphisms of matrix algebras over commutative rings, Linear Algebra Appl., 31, 215-231, (1980) · Zbl 0434.16015 [10] Izumi, M., The K-theory of the flip automorphisms, preprint [11] Jung, H. W.E., Über ganze birationale transformationen der ebene, J. Reine Angew. Math., 184, 161-174, (1942) · JFM 68.0382.01 [12] Kovacs, A., Homomorphisms of matrix rings into matrix rings, Pacific J. Math., 49, 161-170, (1973) · Zbl 0275.16019 [13] Lam, T. Y., A first course in noncommutative rings, Graduate Texts in Mathematics, vol. 131, (2001), Springer · Zbl 0980.16001 [14] Montgomery, S., Von Neumann finiteness of tensor products of algebras, Comm. Algebra, 11, 595-610, (1983) · Zbl 0488.16015 [15] Quillen, D., Projective modules over polynomial rings, Invent. Math., 36, 167-171, (1976) · Zbl 0337.13011 [16] Rosenberg, A.; Zelinsky, D., Automorphisms of separable algebras, Pacific J. Math., 11, 1109-1117, (1961) · Zbl 0116.02501 [17] Sakai, S., Automorphisms and tensor products of operator algebras, Amer. J. Math., 97, 889-896, (1975) · Zbl 0321.46052 [18] Suslin, A., Projective modules over polynomial rings are free, Dokl. Akad. Nauk SSSR, 229, 5, 1063-1066, (1976), (Russian) [19] Shestakov, I. P.; Umirbaev, U. U., The Nagata automorphism is wild, Proc. Natl. Acad. Sci. USA, 100, 12561-12563, (2003) · Zbl 1065.13010
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