linguistic:paraphrasing, linguistic:voice_change
Tensor products of modules over a ring

The tensor product of two modules A and B over a commutative ring R is defined in exactly the same way as the tensor product of vector spaces over a field: A ⊗ R B:= F (A × B) / G, {\displaystyle A\otimes _{R}B:=F(A\times B)/G,} where now F (A × B) {\displaystyle F(A\times B)} is the free R -module generated by the cartesian product and G is the R -module generated by these relations.

More generally, the tensor product can be defined even if the ring is non-commutative. In this case A has to be a right- R -module and B is a left- R -module, and instead of the last two relations above, the relation: (a r, b) ∼ (a, r b) {\displaystyle (ar,b)\sim (a,rb)} is imposed. If R is non-commutative, this is no longer an R -module, but just an abelian group.

The universal property also carries over, slightly modified: the map φ: A × B → A ⊗ R B {\displaystyle \varphi:A\times B\to A\otimes _{R}B} defined by (a, b) ↦ a ⊗ b {\displaystyle (a,b)\mapsto a\otimes b} is a middle linear map (referred to as "the canonical middle linear map"); that is, it satisfies: φ (a + a ′, b) = φ (a, b) + φ (a ′, b) φ (a, b + b ′) = φ (a, b) + φ (a, b ′) φ (a r, b) = φ (a, r b) {\displaystyle {\begin{aligned}\varphi (a+a',b)&=\varphi (a,b)+\varphi (a',b)\\\varphi (a,b+b')&=\varphi (a,b)+\varphi (a,b')\\\varphi (ar,b)&=\varphi (a,rb)\end{aligned}}}

The first two properties make φ a bilinear map of the abelian group A × B {\displaystyle A\times B}. For any middle linear map ψ {\displaystyle \psi } of A × B {\displaystyle A\times B}, a unique group homomorphism f of A ⊗ R B {\displaystyle A\otimes _{R}B} satisfies ψ = f ∘ φ {\displaystyle \psi =f\circ \varphi }, and this property determines φ {\displaystyle \varphi } within group isomorphism. See the main article for details.