Home » Algebra: Module Theory

Tensor product preserves direct sums

Tensoring with a fixed module distributes over arbitrary direct sums.
Tensor product preserves 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. The canonical map

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)

induced by the inclusions Ni→⨁i∈INiN_i\to\bigoplus_{i\in I}N_i is an isomorphism. Likewise, for a family of right RR-modules {Mi}i∈I\{M_i\}_{i\in I} and a left RR-module NN there is a canonical isomorphism

(⨁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 .