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TakuyaMurata (talk | contribs) →Properties: three properties include symmetry |
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Let ''R'' be a commutative ring, and ''M'', ''N'' and ''P'' be ''R''-modules. Then
; 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>
; Symmetry : <math display=block>M \otimes_R N = N \otimes_R M.</math> In fact, for any permutation ''σ'' of the set {1, ..., ''n''}, there is a unique isomorphism: <math display="block">\begin{cases} M_1 \otimes_R \cdots \otimes_R M_n \longrightarrow M_{\sigma(1)} \otimes_R \cdots \otimes_R M_{\sigma(n)} \\ x_1 \otimes \cdots \otimes x_n \longmapsto x_{\sigma(1)} \otimes \cdots \otimes x_{\sigma(n)} \end{cases}</math>
: The first three properties (plus identities on morphisms) say that the category of ''R''-modules, with ''R'' commutative, forms a [[symmetric monoidal category]].
; Distribution over [[direct sum]]s : <math display=block>M \otimes_R (N \oplus P) = (M \otimes_R N) \oplus (M \otimes_R P).</math> In fact, <math display="block">M \otimes_R \left (\bigoplus\nolimits_{i \in I} N_i \right ) = \bigoplus\nolimits_{i \in I} \left ( M \otimes_R N_i \right ),</math> for an [[index set]] ''I'' of arbitrary [[cardinality]]. Since finite products coincide with finite direct sums, this imples:
*; Distribution over finite products : For any finitely many <math>N_i</math>, <math display="block">M \otimes_R \prod_{i = 1}^n N_i = \prod_{i = 1}^nM \otimes_R N_i.</math>
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