information:classification, information:filtering, structural:reformatting
Tensor product of modules over a non-commutative ring

Let A be a right R -module and B be a left R -module. Then the tensor product of A and B is an abelian group defined by: A ⊗ R B:= F (A × B) / G {\displaystyle A\otimes _{R}B:=F(A\times B)/G} where F (A × B) {\displaystyle F(A\times B)} is a free abelian group over A × B {\displaystyle A\times B} and G is the subgroup of F (A × B) {\displaystyle F(A\times B)} generated by relations: ∀ a, a 1, a 2 ∈ A, ∀ b, b 1, b 2 ∈ B, for all r ∈ R: (a 1, b) + (a 2, b) − (a 1 + a 2, b), (a, b 1) + (a, b 2) − (a, b 1 + b 2), (a r, b) − (a, r b). {\displaystyle {\begin{aligned}&\forall a,a_{1},a_{2}\in A,\forall b,b_{1},b_{2}\in B,{\text{ for all }}r\in R:\\&(a_{1},b)+(a_{2},b)-(a_{1}+a_{2},b),\\&(a,b_{1})+(a,b_{2})-(a,b_{1}+b_{2}),\\&(ar,b)-(a,rb).\\\end{aligned}}}

The universal property can be stated as follows. Let G be an abelian group with a map q: A × B → G {\displaystyle q:A\times B\to G} that is bilinear, in the sense that: q (a 1 + a 2, b) = q (a 1, b) + q (a 2, b), q (a, b 1 + b 2) = q (a, b 1) + q (a, b 2), q (a r, b) = q (a, r b). {\displaystyle {\begin{aligned}q(a_{1}+a_{2},b)&=q(a_{1},b)+q(a_{2},b),\\q(a,b_{1}+b_{2})&=q(a,b_{1})+q(a,b_{2}),\\q(ar,b)&=q(a,rb).\end{aligned}}}

Then there is a unique map q ¯: A ⊗ B → G {\displaystyle {\overline {q}}:A\otimes B\to G} such that q ¯ (a ⊗ b) = q (a, b) {\displaystyle {\overline {q}}(a\otimes b)=q(a,b)} for all a ∈ A {\displaystyle a\in A} and b ∈ B {\displaystyle b\in B}.

Furthermore, we can give A ⊗ R B {\displaystyle A\otimes _{R}B} a module structure under some extra conditions: