Definition
Let M,N and R be as in the previous section. The tensor product over R
is an abelian group together with a bilinear map (in the sense defined above)
which is universal in the following sense:
- For every abelian group Z and every bilinear map
- there is a unique group homomorphism
- such that
As with all universal properties, the above property defines the tensor product uniquely up to a unique isomorphism: any other object and bilinear map with the same properties will be isomorphic to M ⊗R N and ⊗. The definition does not prove the existence of M ⊗R N; see below for a construction.
The tensor product can also be defined as a representing object for the functor Z → BilinR(M,N;Z). This is equivalent to the universal mapping property given above.
Strictly speaking, the ring used to form the tensor should be indicated: most modules can be considered as modules over several different rings or over the same ring with a different actions of the ring on the module elements. For example, it can be shown that R ⊗R R and R ⊗Z R are completely different from each other. However in practice, whenever the ring is clear from context, the subscript denoting the ring may be dropped.
Read more about this topic: Tensor Product Of Modules
Famous quotes containing the word definition:
“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)
“... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
—Adrienne Rich (b. 1929)