Czechoslovak Mathematical Journal, Vol. 71, No. 1, pp. 137-154, 2021
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
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