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:
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (1839–1894)
“One definition of man is “an intelligence served by organs.””
—Ralph Waldo Emerson (1803–1882)
“... 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)