A torsionless module is one for which the canonical homomorphism is injective.
2.
A reflexive module is one for which the canonical homomorphism is an isomorphism.
3.
There is always a canonical homomorphism from " E " to its second dual.
4.
Is commonly referred to as the natural homomorphism or "'canonical homomorphism " '.
5.
In general, " E " is called a reflexive module if the canonical homomorphism is an isomorphism.
6.
Every module has a canonical homomorphism to the dual of its dual ( called the "'double dual "').
7.
However, on a more general class of domains including Noetherian domains and Krull domains the ideal class group is constructed in a different way, and there is a canonical homomorphism
8.
We have seen above that there is a canonical homomorphism ? from the group of reals to the outer automorphism group of a von Neumann algebra, given by modular automorphisms.
9.
The above diagram includes the two cases you mentioned, as is demonstrated by the following special commutative diagrams ( let h _ 1 : A \ to A / N and h _ 2 : B \ to B / f ( N ) be the respective canonical homomorphisms ):
10.
Sometimes, filtrations are supposed to satisfy the additional requirement that the union of the S _ i be the whole S, or ( in more general cases, when the notion of union does not make sense ) that the canonical homomorphism from the direct limit of the S _ i to S is an isomorphism.
