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Tensor product preserves direct sums

Tensoring with a fixed module distributes over arbitrary direct sums.

Tensor product preserves direct sums. Let MM be a right RR-module and {Ni}i∈I\{N_i\}_{i\in I} a family of left RR-modules. There is a canonical isomorphism

M⊗R(⨁i∈INi)⟶⨁i∈I(M⊗RNi)M\otimes_R\Bigl(\bigoplus_{i\in I}N_i\Bigr)\longrightarrow \bigoplus_{i\in I}(M\otimes_R N_i)

that sends m⊗(ni)i∈Im\otimes(n_i)_{i\in I} to (m⊗ni)i∈I(m\otimes n_i)_{i\in I}. Likewise, for a family of right RR-modules {Mi}i∈I\{M_i\}_{i\in I} and a left RR-module NN,

(⨁i∈IMi)⊗RN≅⨁i∈I(Mi⊗RN).\Bigl(\bigoplus_{i\in I} M_i\Bigr)\otimes_R N \cong \bigoplus_{i\in I}(M_i\otimes_R N).

This is a basic compatibility of the with the , and can be viewed as a special case of the .