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 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.

Read more about this topic:  Tensor Product Of Modules

Other articles related to "additional structure, structure, structures, additional":

Cobordism - Definition - Variants
... In many situations, the manifolds in question are oriented, or carry some other additional structure referred to as G-structure ... rise to "oriented cobordism" and "cobordism with G-structure", respectively ... When there is additional structure, the notion of cobordism must be formulated more precisely a -structure on restricts to a -structure on and ...
Classification Of Manifolds - Main Themes - Different Categories and Additional Structure
... nor onto these failures are generally referred to in terms of "structure", as follows ... manifold that is in the image of is said to "admit a differentiable structure", and the fiber over a given topological manifold is "the different differentiable structures on the given topological manifold" ... are Which manifolds of a given type admit an additional structure? If it admits an additional structure, how many does it admit? More precisely, what is the structure of the set of additional ...
Finite Topological Space - Properties - Additional Structure
... Likewise, a topological space is uniformizable if and only if it is R0 ... The uniform structure will be the pseudometric uniformity induced by the above pseudometric ...
Polyhedral Space - Additional Structure
... holonomies also preserve a symplectic form, together with a complex structure on this polyhedral space (manifold) with the singularities removed ...

Famous quotes containing the words structure and/or additional:

    What is the most rigorous law of our being? Growth. No smallest atom of our moral, mental, or physical structure can stand still a year. It grows—it must grow; nothing can prevent it.
    Mark Twain [Samuel Langhorne Clemens] (1835–1910)

    When I turned into a parent, I experienced a real and total personality change that slowly shifted back to the “normal” me, yet has not completely vanished. I believe the two levels are now superimposed, with an additional sprinkling of mortality intimations.
    Sonia Taitz (20th century)