# Tensor Product of Modules - 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 MRN can be made into a left S-module using the obvious operation s(mn)=(smn). Similarly, if N is an (R,T)-bimodule, then MRN is a right T-module using the operation (mn)t=(mnt). If M and N each have bimodule structures as above, then MRN 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 MRN 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(mn) − m ⊗ (r·n), or equivalently the elements (m·r) ⊗ n − r(mn).

If {mi}iI and {nj}jJ are generating sets for M and N, respectively, then {minj}iI,jJ will be a generating set for MN. Because the tensor functor MR- 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 MF N.

If S and T are commutative R-algebras, then SR T will be a commutative R-algebra as well, with the multiplication map defined by (m1m2)(n1n2) = (m1n1m2n2) 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 MZ N.

If S1MR is an S1-R-bimodule, then there is a unique left S1-module structure on MN which is compatible with the tensor map ⊗:M×NMRN. Similarly, if RNS2 is an R-S2-bimodule, then there is a unique right S2-module structure on MRN 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

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.

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