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 to, relationship and/or flat:
“... the Wall became a magnet for citizens of every generation, class, race, and relationship to the war perhaps because it is the only great public monument that allows the anesthetized holes in the heart to fill with a truly national grief.”
—Adrienne Rich (b. 1929)
“Christianity as an organized religion has not always had a harmonious relationship with the family. Unlike Judaism, it kept almost no rituals that took place in private homes. The esteem that monasticism and priestly celibacy enjoyed implied a denigration of marriage and parenthood.”
—Beatrice Gottlieb, U.S. historian. The Family in the Western World from the Black Death to the Industrial Age, ch. 12, Oxford University Press (1993)
“But pardon, gentles all,
The flat unraised spirits that hath dared
On this unworthy scaffold to bring forth
So great an object.”
—William Shakespeare (1564–1616)