central simple algebra - Weblio 英和・和英辞典 (original) (raw)

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意味・対訳

central simple algebraの意味・対訳は、数学の特に環論において、体 K 上の中心的単純多元環（ちゅうしんてきたんじゅんかん、英: central simple algebra; CSA）とは、与えられた K 上の階数（ベクトル空間としての次元）が有限な結合多元環 A であって、環として単純で、その中心がちょうど K となっているようなものをいう。、などです。

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Central simple algebra

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central simple algebra

出典:『Wiktionary』 (2025/10/26 23:24 UTC 版)

名詞

central simple algebra (plural central simple algebras)

(algebra, ring theory) A finite-dimensional associative algebra over some field K that is a simple algebra and whose centre is exactly K.
The complex numbers C {\displaystyle \mathbb {C} } $\mathbb {C}$ form a central simple algebra over themselves, but not over the real numbers R {\displaystyle \mathbb {R} } $\mathbb {R}$ (the centre of C {\displaystyle \mathbb {C} } $\mathbb {C}$ is all of C {\displaystyle \mathbb {C} } $\mathbb {C}$ , not just R {\displaystyle \mathbb {R} } $\mathbb {R}$ ). The quaternions H {\displaystyle \mathbb {H} } $\mathbb {H}$ form a 4-dimensional central simple algebra over R {\displaystyle \mathbb {R} } $\mathbb {R}$ .
The concept of central simple algebra over a field K represents a noncommutative analogue to that of extension field over K. In both cases, the object has no nontrivial two-sided ideals and has a distinguished field in its centre, although a central simple algebra need not be commutative and need not have inverses (does not have be a division algebra).
- 1987, Gregory Karpilovsky, The Algebraic Structure of Crossed Products‎, Elsevier (North-Holland), page 151:
  This crossed product E α G {\displaystyle E^{\alpha }G} $E^{\alpha }G$ was introduced by Noether and played a significant role in the classical theory of central simple algebras.
- 2007, Falko Lorenz, Algebra: Volume II: Fields with Structure, Algebras and Advanced Topics, Springer, page 151:
- 2014, Jörg Jahnel, Brauer Groups, Tamagawa Measures, and Rational Points on Algebraic Varieties, American Mathematical Society, page 84:
  Let A 1 , A 2 {\displaystyle A_{1},A_{2}} $A_{1},A_{2}$ be central simple algebras over a field K {\displaystyle K} $K$ . Then A 1 ⊗ K A 2 {\displaystyle A_{1}\otimes _{K}A_{2}} $A_{1}\otimes _{K}A_{2}$ can be shown to be a central simple algebra over K {\displaystyle K} $K$ . Further, if A {\displaystyle A} $A$ is a central simple algebra over a field K {\displaystyle K} $K$ , then A ⊗ K A op ≅ Aut K - V e c t ⁡ ( A ) {\displaystyle A\otimes _{K}A^{\operatorname {op} }\cong \operatorname {Aut} _{K\operatorname {-Vect} }(A)} $A\otimes _{K}A^{\operatorname {op} }\cong \operatorname {Aut} _{K\operatorname {-Vect} }(A)$ . I.e., it is isomorphic to a matrix algebra.

central simple algebra - Weblio 英和・和英辞典 (original) (raw)

Central simple algebra

central simple algebra

名詞

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