English Dictionary

algebra over a ring

名詞

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

日本語の意味

環上代数とは、ある環 R 上の加群（module）の上に定義された代数的構造であり、この加群に対して、加群の元同士の二項演算（積）が定義されています。この積は加群の加法およびスカラー乗法に関して双線形となるため、分配法則が成立し、結果としてその加群は環としての性質も備えることになります。

An algebra over a ring is a module endowed with a bilinear multiplication between module elements that distributes over addition and is compatible with scalar multiplication.

環上の代数は、係数が環である加群に加群元同士の双線形な積が備わったもので、その積は加法に関して分配しスカラー倍と整合します。

意味（1）

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

algebras over rings （ plural ）