In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right-module and a left-module over any ring, with result an abelian group. Tensor products are important in areas of abstract algebra, homological algebra, algebraic topology, algebraic geometry, operator algebras and noncommutative geometry. The universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. The tensor product of an algebra and a module can be used for extension of scalars. For a commutative ring, the tensor product of modules can be iterated to form the tensor algebra of a module, allowing one to define multiplication in the module in a universal way.
Balanced product
For a ring R, a right R-module M, a left R-module N, and an abelian group G, a map φ: M × N → G is said to be R-balanced, R-middle-linear or an R-balanced product if for all m, m′ in M, n, n′ in N, and r in R the following hold:
φ ( m , n + n ′ ) = φ ( m , n ) + φ ( m , n ′ ) Dl φ φ ( m + m ′ , n ) = φ ( m , n ) + φ ( m ′ , n ) Dr φ φ ( m ⋅ r , n ) = φ ( m , r ⋅ n ) A φ {\displaystyle {\begin{aligned}\varphi (m,n+n')&=\varphi (m,n)+\varphi (m,n')&&{\text{Dl}}_{\varphi }\\\varphi (m+m',n)&=\varphi (m,n)+\varphi (m',n)&&{\text{Dr}}_{\varphi }\\\varphi (m\cdot r,n)&=\varphi (m,r\cdot n)&&{\text{A}}_{\varphi }\\\end{aligned}}}
The set of all such balanced products over R from M × N to G is denoted by LR(M, N; G). If φ, ψ are balanced products, then each of the operations φ + ψ and −φ defined pointwise is a balanced product. This turns the set LR(M, N; G) into an abelian group. For M and N fixed, the map G ↦ LR(M, N; G) is a functor from the category of abelian groups to itself. The morphism part is given by mapping a group homomorphism g : G → G′ to the function φ ↦ g ∘ φ, which goes from LR(M, N; G) to LR(M, N; G′).
Remarks
Properties (Dl) and (Dr) express biadditivity of φ, which may be regarded as distributivity of φ over addition. Property (A) resembles some associative property of φ. Every ring R is an R-bimodule. So the ring multiplication (r, r′) ↦ r ⋅ r′ in R is an R-balanced product R × R → R.
Definition For a ring R, a right R-module M, a left R-module N, the tensor product over R
M ⊗ R N {\displaystyle M\otimes _{R}N}
is an abelian group together with a balanced product (as defined above)
⊗ : M × N → M ⊗ R N {\displaystyle \otimes :M\times N\to M\otimes _{R}N}
which is universal in the following sense:
For every abelian group G and every balanced product f : M × N → G {\displaystyle f:M\times N\to G} there is a unique group homomorphism f ~ : M ⊗ R N → G {\displaystyle {\tilde {f}}:M\otimes _{R}N\to G} such that f ~ ∘ ⊗ = f . {\displaystyle {\tilde {f}}\circ \otimes =f.}
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