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Tensor product of modules

April 16, 2012

Like the free group and the free module, the tensor product of two R-modules is another construction satisfying an interesting universal property, but with a richer structure than these two. Namely, it is (when possible) an R-module, denoted M \otimes _R N, such that every bilinear map from M \times N to another R-module uniquely factors through M \otimes _R N.

Given a unitary ring R, a right R-module M and a left R-module N, we will first construct the tensor products of M and N as an abelian group. We can then, when possible, give it a module structure.

Definition: The tensor product of M and N over the ring R, noted M \otimes _R N or M \otimes N, is the quotient of the free \mathbf{Z}-module (or equivalently the free abelian group) over M \times N by the subgroup generated by all elements of the form (\dagger)

(m_1+m_2,n) - (m_1,n) - (m_2,n);


(mr,n) - (m,rn).

for any m, m_1, m_2 \in M, n,n_1, n_2 \in N and r \in R. Elements of M \otimes N are called tensors and the cosets, denoted m \otimes n, represented by (m,n), are called simple tensors.

Note the similarity between this definition and that of the free group or free module: we took all couples in M \times N and quotiented out by the minimal relations giving us the most general module possessing the properties we wanted, namely

(m_1 + m_2) \otimes n = m_1 \otimes n + m_2 \otimes n ;

m \otimes (n_1 + n_2) = m \otimes n_1 + m \otimes n_2 ;

mr \otimes n = m \otimes rn.

for all m, m_1, m_2 \in M, n, n_1, n_2 \in N, r \in R. (Note the similarity between these and the definition of a bilinear map.) It is easy to see that every tensor is a finite sum of simple tensors and that M \otimes N is an abelian group (for addition of tensors).

Definition: For S and R two rings, an (S,R)bimodule is a module which is a left S-module and a right R-module.

Suppose M is an (S,R)-bimodule, then M \otimes _R N can be given a left S-module structure by letting

s.\left(\displaystyle\sum_{finite}m_i \otimes n_i \right) = \displaystyle\sum_{finite}(s.m_i) \otimes n_i.

An important remark is that for a commutative ring R and a R-module M, letting the left and right actions of R on M be equal by r.m := m.r gives M a natural (R,R)-bimodule structure and thus gives M \otimes _R N a left R-module structure. To simplify matters, I will therefore assume in the rest of this post that R is commutative and that M has this natural bimodule structure.

Theorem (Universal property): Let R be a commutative and unitary ring, M, N and L three R-modules and \varphi : M \times N \to L a bilinear map. Then there exists a unique R-module homomorphism \phi : M \otimes _R N \to L such that \phi(m\otimes n) = \varphi (m,n). ie such that this diagram commutes :

(Where \iota(m,n) = m \otimes n.)

Proof: By the universal property of free abelian groups (identifying abelian groups with \textbf{Z}-modules) the map \varphi defines a unique \textbf{Z}-module homomorphism \tilde\varphi : F(M \times N) \to L from the free \textbf{Z}-module on M \times N to L such that \tilde\varphi(m,n) = \varphi(m,n). Since \varphi is bilinear, \tilde\varphi sends every element of the form (\dagger) to 0 \in L so \ker \tilde\varphi contains the subgroup by which we quotiented F(M \times N) to obtain M \otimes N at the beginning of this post. Letting \phi : M \otimes N \to L be \phi(m\otimes n) = \tilde\varphi(m,n) thus gives us a well-defined group homomorphism \phi : M \otimes N \to L such that \varphi = \phi \circ \iota. Moreover, we have

\phi(r(m\otimes n)) = \varphi(rm,n) = r \varphi(m,n) = r\phi(m \otimes n)

so \phi is also an R-module homomorphism. Since \iota(M \times N) generates M \otimes N, (every tensor is a finite sum of simple tensors) we can conclude by linearity that \phi is the unique R-module homomorphism satisfying these properties.


Note that for every R-module homomorphism \phi : M \otimes _R N \to L, the map \varphi(m,n) := \phi(m\otimes n) is an R-bilinear map. So by the last theorem, there is a bijection between the sets \textbf{Bilin}_R(M \times N, L) and \textbf{Hom}(M \otimes _R N, L).

Reference: D. S. Dummit, R. M. Foote, Abstract Algebra, 3rd ed.


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