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In mathematics, an algebra over a field is a vector space (a module over a field) equipped with a bilinear product. Thus, an algebra over a field is a set, together with operations of multiplication, addition, and scalar multiplication by elements of the underlying field, that satisfy the axioms implied by "vector space" and "bilinear".〔See also 〕 The multiplication operation in an algebra may or may not be associative, leading to the notions of associative algebras and nonassociative algebras. Given an integer ''n'', the ring of real square matrices of order ''n'' is an example of an associative algebra over the field of real numbers under matrix addition and matrix multiplication. Euclidean space with multiplication given by the vector cross product is an example of a nonassociative algebra over the field of real numbers. An algebra is unital or unitary if it has an identity element with respect to the multiplication. The ring of real square matrices of order ''n'' forms a unital algebra since the identity matrix of order ''n'' is the identity element with respect to matrix multiplication. It is an example of a unital associative algebra, a (unital) ring that is also a vector space. Many authors use the term ''algebra'' to mean ''associative algebra'', or ''unital associative algebra'', or in some subjects such as algebraic geometry, ''unital associative commutative algebra''. Replacing the field of scalars by a commutative ring leads to the more general notion of an algebra over a ring. Algebras are not to be confused with vector spaces equipped with a bilinear form, like inner product spaces. == Definition and motivation == 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「algebra over a field」の詳細全文を読む スポンサード リンク
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