Relationship To Flat Modules
In general, is a bifunctor which accepts a right and a left R module pair as input, and assigns them to the tensor product in the category of abelian groups.
By fixing a right R module M, a functor arises, and symmetrically a left R module N could be fixed to create a functor . Unlike the Hom bifunctor, the tensor functor is covariant in both inputs.
It can be shown that M⊗- and -⊗N are always right exact functors, but not necessarily left exact. By definition, a module T is a flat module if T⊗- is an exact functor.
If {mi}i∈I and {nj}j∈J are generating sets for M and N, respectively, then {mi⊗nj}i∈I,j∈J will be a generating set for M⊗N. Because the tensor functor M⊗R- sometimes fails to be left exact, this may not be a minimal generating set, even if the original generating sets are minimal.
When the tensor products are taken over a field F so that -⊗- is exact in both positions, and the generating sets are bases of M and N, it is true that indeed forms a basis for M⊗F N.
Read more about this topic: Tensor Product Of Modules
Famous quotes containing the words relationship and/or flat:
“Friendship is by its very nature freer of deceit than any other relationship we can know because it is the bond least affected by striving for power, physical pleasure, or material profit, most liberated from any oath of duty or of constancy.”
—Francine Du Plesssix Gray (20th century)
“It would astonish if not amuse, the older citizens of your County who twelve years ago knew me a stranger, friendless, uneducated, penniless boy, working on a flat boat—at ten dollars per month to learn that I have been put down here as the candidate of pride, wealth, and aristocratic family distinction.”
—Abraham Lincoln (1809–1865)