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TakuyaMurata (talk | contribs) →Modules over commutative rings: the reasoning refers to the previous property |
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=== Modules over commutative rings ===
Let ''R'' be a commutative ring, and ''M'', ''N'' and ''P'' be ''R''-modules. Then (in the below, "=" denotes [[canonical isomorphism]]s; this attitude is permissible since a tensor product is defined only up to unique isomorphisms)
; Identity : <math display=block>R \otimes_R M = M.</math>
; Associativity : <math display=block>(M \otimes_R N) \otimes_R P = M \otimes_R (N \otimes_R P).</math>
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<math display="block">\operatorname{Hom}_S (M \otimes_R S, P) = \operatorname{Hom}_R (M, \operatorname{Res}_R(P)).</math>
This says that the functor <math>-\otimes_R S</math> is a [[left adjoint]] to the forgetful functor {{tmath|1= \operatorname{Res}_R }}, which restricts an ''S''-action to an ''R''-action. Because of this, <math>- \otimes_R S</math> is often called the [[extension of scalars]] from ''R'' to ''S''. In
==== Examples ====
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