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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: Fix a right RR-module MM. Then the functor M⊗R−M\otimes_R - preserves all small colimits of left RR-modules; in particular, for any family {Ni}i∈I\{N_i\}_{i\in I} 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)

is an isomorphism, and for any directed system {Ni}\{N_i\} the canonical map lim→⁡i(M⊗RNi)→M⊗R(lim→⁡iNi)\varinjlim_i(M\otimes_R N_i)\to M\otimes_R(\varinjlim_i N_i) is an isomorphism.

Formally this follows from the (tensoring is a left adjoint), and the direct-sum case recovers for .