Czechoslovak Mathematical Journal, first online, pp. 1-18


A note on Skolem-Noether algebras

Juncheol Han, Tsiu-Kwen Lee, Sangwon Park

Received May 13, 2019.   Published online April 23, 2020.

Abstract:  The paper was motivated by Kovacs' paper (1973), Isaacs' paper (1980) and a recent paper, due to Brešar et al. (2018), concerning Skolem-Noether algebras. Let $K$ be a unital commutative ring, not necessarily a field. Given a unital $K$-algebra $S$, where $K$ is contained in the center of $S$, $n\in\mathbb N$, the goal of this paper is to study the question: when can a homomorphism $\phi\colon{\rm M}_n(K)\to{\rm M}_n(S)$ be extended to an inner automorphism of ${\rm M}_n(S)$? As an application of main results presented in the paper, it is proved that if $S$ is a semilocal algebra with a central separable subalgebra $R$, then any homomorphism from $R$ into $S$ can be extended to an inner automorphism of $S$.
Keywords:  Skolem-Noether algebra; (inner) automorphism; matrix algebra; central simple algebra; central separable algebra; semilocal ring; unique factorization domain (UFD); stably finite ring; Dedekind-finite ring
Classification MSC:  16K20, 16W20, 16S50
DOI:  10.21136/CMJ.2020.0215-19

PDF available at:  Springer   Institute of Mathematics CAS

References:
[1] M. Brešar, C. Hanselka, I. Klep, J. Volčič: Skolem-Noether algebras. J. Algebra 498 (2018), 294-314. DOI 10.1016/j.jalgebra.2017.11.045 | MR 3754416 | Zbl 06834833
[2] F. DeMeyer, E. Ingraham: Separable Algebras over Commutative Rings. Lecture Notes in Mathematics 181, Springer, Berlin (1971). DOI 10.1007/BFb0061226 | MR 0280479 | Zbl 0215.36602
[3] I. N. Herstein: Noncommutative Rings. The Carus Mathematical Monographs 15, Mathematical Association of America, New York (1968). DOI 10.5948/upo9781614440154 | MR 0227205 | Zbl 0177.05801
[4] I. M. Isaacs: Automorphisms of matrix algebras over commutative rings. Linear Algebra Appl. 31 (1980), 215-231. DOI 10.1016/0024-3795(80)90221-9 | MR 0570392 | Zbl 0434.16015
[5] I. Kaplansky: Fields and Rings. Chicago Lectures in Mathematics, The University of Chicago Press, Chicago (1969). MR 0269449 | Zbl 0238.16001
[6] A. Kovacs: Homomorphisms of matrix rings into matrix rings. Pac. J. Math. 49 (1973), 161-170. DOI 10.2140/pjm.1973.49.161 | MR 0447332 | Zbl 0275.16019
[7] T. Y. Lam: A First Course in Noncommutative Rings. Graduate Texts in Mathematics 131, Springer, New York (1991). DOI 10.1007/978-1-4684-0406-7 | MR 1125071 | Zbl 0728.16001
[8] N. H. McCoy: Subdirectly irreducible commutative rings. Duke Math. J. 12 (1945), 381-387. DOI 10.1215/S0012-7094-45-01232-4 | MR 0012266 | Zbl 0060.05901
[9] A. Milinski: Skolem-Noether theorems and coalgebra actions. Commun. Algebra 21 (1993), 3719-3725. DOI 10.1080/00927879308824760 | MR 1231628 | Zbl 0793.16030
[10] A. Rosenberg, D. Zelinsky: Automorphisms of separable algebras. Pac. J. Math. 11 (1961), 1109-1117. DOI 10.2140/pjm.1961.11.1109 | MR 0148709 | Zbl 0116.02501
[11] L. Rowen: Some results on the center of a ring with polynomial identity. Bull. Am. Math. Soc. 79 (1973), 219-223. DOI 10.1090/S0002-9904-1973-13162-3 | MR 0309996 | Zbl 0252.16007
[12] J. B. Srivastava, S. K. Shah: Semilocal and semiregular group rings. Indag. Math. 42 (1980), 347-352. DOI 10.1016/1385-7258(80)90035-9 | MR 0587061 | Zbl 0442.16010

Affiliations:   Juncheol Han, Department of Mathematics Education, Pusan National University, 2, Busandaehak-ro 63beon-gil, Geumjeong-gu, Pusan, 609-735, South Korea, e-mail: jchan@pusan.ac.kr, Tsiu-Kwen Lee (corresponding author), Department of Mathematics, National Taiwan University, Astronomy Mathematics Building 5F, No. 1, Sec. 4, Roosevelt Rd., Taipei 10617, Taiwan, e-mail: tklee@math.ntu.edu.tw, Sangwon Park, Department of Mathematics, Dong-A University, 37 Nakdong-daero 550beon-gil Saha-gu, Pusan, 604-714, South Korea, e-mail: swpark@donga.ac.kr


 
PDF available at: