Tensor Product of Modules - Construction

Construction

The construction of M ⊗ N takes a quotient of a free abelian group with basis the symbols m ⊗ n for m in M and n in N by the subgroup generated by all elements of the form

1. −(m+m′) ⊗ n + m ⊗ n + m′ ⊗ n
2. −m ⊗ (n+n′) + m ⊗ n + m ⊗ n′
3. (m·r) ⊗ n − m ⊗ (r·n)

where m,m′ in M, n,n′ in N, and r in R. The function which takes (m,n) to the coset containing m ⊗ n is bilinear, and the subgroup has been chosen minimally so that this map is bilinear.

The direct product of M and N is rarely isomorphic to the tensor product of M and N. When R is not commutative, then the tensor product requires that M and N be modules on opposite sides, while the direct product requires they be modules on the same side. In all cases the only function from M × N to Z which is both linear and bilinear is the zero map.