Tensor Product of Modules - Several Modules

Several Modules

It is possible to generalize the definition to a tensor product of any number of spaces. For example, the universal property of

M₁ ⊗ M₂ ⊗ M₃

is that each trilinear map on

M₁ × M₂ × M₃ → Z

corresponds to a unique linear map

M₁ ⊗ M₂ ⊗ M₃ → Z.

The binary tensor product is associative: (M₁ ⊗ M₂) ⊗ M₃ is naturally isomorphic to M₁ ⊗ (M₂ ⊗ M₃). The tensor product of three modules defined by the universal property of trilinear maps is isomorphic to both of these iterated tensor products.