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In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field. The addition and multiplication operations together give ''A'' the structure of a ring; the addition and scalar multiplication operations together give ''A'' the structure of a vector space over ''K''. In this article we will also use the term ''K''-algebra to mean an associative algebra over the field ''K''. A standard first example of a ''K''-algebra is a ring of square matrices over a field ''K'', with the usual matrix multiplication. In this article associative algebras are assumed to have a multiplicative unit, denoted 1; they are sometimes called unital associative algebras for clarification. In some areas of mathematics this assumption is not made, and we will call such structures non-unital associative algebras. We will also assume that all rings are unital, and all ring homomorphisms are unital. Many authors consider the more general concept of an associative algebra over a commutative ring ''R'', instead of a field: An ''R''-algebra is an ''R''-module with an associative ''R''-bilinear binary operation, which also contains a multiplicative identity. For examples of this concept, if ''S'' is any ring with center ''C'', then ''S'' is an associative ''C''-algebra. == Definition == Let ''R'' be a fixed commutative ring (so ''R'' could be a field). An associative ''R''-algebra (or more simply, an ''R''-algebra) is an additive abelian group ''A'' which has the structure of both a ring and an ''R''-module in such a way that the scalar multiplication satisfies : for all ''r'' ∈ ''R'' and ''x'', ''y'' ∈ ''A''. Furthermore, ''A'' is assumed to be unital, which is to say it contains an element 1 such that : for all ''x'' ∈ ''A''. Note that such an element 1 must be unique. In other words, ''A'' is an ''R''-module together with (1) an ''R''-bilinear map ''A'' × ''A'' → ''A'', called the multiplication, and (2) the multiplicative identity, such that the multiplication is associative: : for all ''x'', ''y'', and ''z'' in ''A''. (Technical note: the multiplicative identity is an additional datum, while associativity is a property. By the uniqueness of the multiplicative identity, "unitarity" is often treated like a property.) If one drops the requirement for the associativity, then one obtains a non-associative algebra. If ''A'' itself is commutative (as a ring) then it is called a commutative ''R''-algebra. 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「Associative algebra」の詳細全文を読む スポンサード リンク
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