Tensor Product of Modules - Multilinear Mappings

Multilinear Mappings

For a ring R, a right R-module M_R, a left R-module _RN, and an abelian group Z, a bilinear map or balanced product from M × N to Z is a function φ: M × N → Z such that for all m,m′ in M, n,n′ in N, and r in R:

1. φ(m+m′,n) = φ(m,n) + φ(m′,n)
2. φ(m,n+n′) = φ(m,n) + φ(m,n′)
3. φ(m·r,n) = φ(m,r·n)

The set of all such bilinear maps from M × N to Z is denoted by Bilin(M,N;Z).

Property 3 differs slightly from the definition for vector spaces. This is necessary because Z is only assumed to be an abelian group, so r·φ(m,n) would not make sense.

If φ, ψ are bilinear maps, then φ + ψ is a bilinear map, and -φ is a bilinear map, when these operations are defined pointwise. This turns the set Bilin(M,N;Z) into an abelian group. The neutral element is the zero mapping.

For M and N fixed, the map Z ↦ Bilin(M,N;Z) is a functor from the category of abelian groups to the category of sets. The morphism part is given by mapping a group homomorphism g : Z → Z′ to the function, which goes from Bilin(M,N;Z) to Bilin(M,N;Z′).