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The tensor product represents balanced bilinear maps out of a pair of modules.
Let be a right -module and a left -module. There is an abelian group and a biadditive map
satisfying . For every abelian group and every biadditive map satisfying , there is a unique group homomorphism such that .
This is the standard representing property of the tensor product, packaging bilinear maps into a universal object; compare the tensor product universal property.
The tensor product of modules of a right -module and a left -module (for a ring ) is an abelian group equipped with a canonical balanced bilinear map , , satisfying the universal property: every balanced bilinear map out of factors uniquely through a homomorphism out of .
This construction is functorial in both variables and is central for extension of scalars and for measuring non-exactness via derived functors. Over commutative rings it specializes to the tensor product of left modules.
A bilinear map between -modules (for a commutative ring ) is a function from a cartesian product of -modules such that, for each fixed argument, the resulting map is -linear in the other:
Let be a ring, a right -module, and a left -module. A pair has the universal property of the tensor product if is an abelian group, is biadditive and balanced,
and every biadditive balanced map to an abelian group factors uniquely through a homomorphism .
As a diagram, the triangle commutes, :
When such exists, it is unique up to unique isomorphism; one writes and , producing the tensor product. The universal property is the mechanism that turns bilinear constructions into linear ones (i.e. module homomorphisms, when the target has compatible structure).