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In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps (module homomorphisms). The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right-module and a left-module over any ring, with result an abelian group. Tensor products are important in areas of abstract algebra, homological algebra, algebraic topology and algebraic geometry. The universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. It allows the study of bilinear or multilinear operations via linear operations. The tensor product of an algebra and a module can be used for extension of scalars. For a commutative ring, the tensor product of modules can be iterated to form the tensor algebra of a module, allowing one to define multiplication in the module in a universal way. ==Balanced product== For a ring ''R'', a right ''R''-module ''M'', a left ''R''-module ''N'', and an abelian group ''G'', a map is said to be ''R''-balanced, ''R''-middle-linear or an ''R''-balanced product if for all ''m'', ''m''′ in ''M'', ''n'', ''n''′ in ''N'', and ''r'' in ''R'' the following hold: : The set of all such balanced products over ''R'' from to ''G'' is denoted by . If ''φ'', ''ψ'' are balanced products, then the operations and −''φ'' defined pointwise are each a balanced product. This turns the set into an abelian group. For ''M'' and ''N'' fixed, the map is a functor from the category of abelian groups to the category of sets. The morphism part is given by mapping a group homomorphism to the function , which goes from to . ;Remarks: #Property (Dl) states the left and property (Dr) the right distributivity of ''φ'' over addition. #Property (A) resembles some associative property of ''φ''. #Every ring ''R'' is an ''R''-''R''-bimodule. So the ring multiplication in ''R'' is an ''R''-balanced product . 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 modules」の詳細全文を読む スポンサード リンク
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