Tensor Product of Modules - Additional Structure

Additional Structure

The tensor product, as defined, is an abelian group, but in general, it does not immediately have an R-module structure. However, if M is an (S,R)-bimodule, then M⊗_RN can be made into a left S-module using the obvious operation s(m⊗n)=(sm⊗n). Similarly, if N is an (R,T)-bimodule, then M⊗_RN is a right T-module using the operation (m⊗n)t=(m⊗nt). If M and N each have bimodule structures as above, then M⊗_RN is an (S,T)-bimodule. In the case where R is a commutative ring, all of its modules can be thought of as (R,R)-bimodules, and then M⊗_RN can be made into an R-module as described. In the construction of the tensor product over a commutative ring R, the multiplication operation can either be defined a posteriori as just described, or can be built in from the start by forming the quotient of a free R-module by the submodule generated by the elements given above for the general construction, augmented by the elements r(m ⊗ n) − m ⊗ (r·n), or equivalently the elements (m·r) ⊗ n − r(m ⊗ n).

If {m_i}_i∈I and {n_j}_j∈J are generating sets for M and N, respectively, then {m_i⊗n_j}_i∈I,j∈J will be a generating set for M⊗N. Because the tensor functor M⊗_R- is right exact, but sometimes not left exact, this may not be a minimal generating set, even if the original generating sets are minimal. If M is a flat module, the functor is exact by the very definition of a flat module. If the tensor products are taken over a field F, we are in the case of vector spaces as above. Since all F modules are flat, the bifunctor is exact in both positions, and the two given generating sets are bases, then indeed forms a basis for M ⊗_F N.

If S and T are commutative R-algebras, then S ⊗_R T will be a commutative R-algebra as well, with the multiplication map defined by (m₁ ⊗ m₂)(n₁ ⊗ n₂) = (m₁n₁ ⊗ m₂n₂) and extended by linearity. In this setting, the tensor product become a fibered coproduct in the category of R-algebras. Note that any ring is a Z-algebra, so we may always take M ⊗_Z N.

If _S1M_R is an S₁-R-bimodule, then there is a unique left S₁-module structure on M⊗N which is compatible with the tensor map ⊗:M×N→M⊗_RN. Similarly, if _RN_S2 is an R-S₂-bimodule, then there is a unique right S₂-module structure on M⊗_RN which is compatible with the tensor map.

If M and N are both R-modules over a commutative ring, then their tensor product is again an R-module. If R is a ring, _RM is a left R-module, and the commutator

rs − sr

of any two elements r and s of R is in the annihilator of M, then we can make M into a right R module by setting

mr = rm.

The action of R on M factors through an action of a quotient commutative ring. In this case the tensor product of M with itself over R is again an R-module. This is a very common technique in commutative algebra.