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In mathematics, the tensor product of two ''R''-algebras is also an ''R''-algebra. This gives us a tensor product of algebras. The special case ''R'' = Z gives us a tensor product of rings, since rings may be regarded as Z-algebras. == Definition == Let ''R'' be a commutative ring and let ''A'' and ''B'' be ''R''-algebras. Since ''A'' and ''B'' may both be regarded as ''R''-modules, we may form their tensor product : which is also an ''R''-module. We can give the tensor product the structure of an algebra by defining the product on elements of the form by〔Kassel (1995), (p. 32 ).〕 : and then extending by linearity to all of . This product is ''R''-bilinear, associative, and unital with an identity element given by ,〔Kassel (1995), (p. 32 ).〕 where 1''A'' and 1''B'' are the identities of ''A'' and ''B''. If ''A'' and ''B'' are both commutative then the tensor product is commutative as well. The tensor product turns the category of all ''R''-algebras into a symmetric monoidal category. 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「tensor product of algebras」の詳細全文を読む スポンサード リンク
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