English Dictionary

algebra over a field

名詞

(algebra) A vector space (over some field) with an additional binary operation, a vector-valued product between vectors, which is bilinear over vector addition and scalar multiplication. (N.B.: such bilinearity implies distributivity of the vector multiplication with respect to the vector addition, which means that such a vector space is also a ring.)

日本語の意味

体上代数とは、ある体上で定義されたベクトル空間において、ベクトル間の積（すなわち二項演算）が加法およびスカラー倍に対して両線形となる構造を持つものを指す。 / この両線形性により、ベクトル積は分配法則を満たし、その結果、このベクトル空間は環の構造も備える。

To study the representations of an algebra over a field, one must understand both its vector space structure and its bilinear multiplication.

体上の代数の表現を研究するには、そのベクトル空間としての構造と双線形な積の両方を理解する必要がある。

意味（1）

(algebra) A vector space (over some field) with an additional binary operation, a vector-valued product between vectors, which is bilinear over vector addition and scalar multiplication. (N.B.: such bilinearity implies distributivity of the vector multiplication with respect to the vector addition, which means that such a vector space is also a ring.)

algebras over a field （ plural ）