Tensor commutes with direct limits and sums

Tensoring is a left adjoint, hence it preserves direct sums and filtered colimits.

Tensor commutes with direct limits and sums

Tensor commutes with direct limits and sums: Fix a right $R$ -module $M$ . Then the functor $M\otimes_R -$ preserves all small colimits of left $R$ -modules; in particular, for any family $\{N_i\}_{i\in I}$ the canonical map

M\otimes_R\Bigl(\bigoplus_{i\in I}N_i\Bigr)\longrightarrow \bigoplus_{i\in I}(M\otimes_R N_i)

is an isomorphism, and for any directed system $\{N_i\}$ the canonical map $\varinjlim_i(M\otimes_R N_i)\to M\otimes_R(\varinjlim_i N_i)$ is an isomorphism.

Formally this follows from the Tensor–Hom adjunction (tensoring is a left adjoint), and the direct-sum case recovers tensor product preserves direct sums for direct sums .