Properties
- The injective hull of M is unique up to isomorphisms which are the identity on M, however the isomorphism is not necessarily unique. This is because the injective hull's map extension property is not a full fledged universal property. Because of this uniqueness, the hull can be denoted as E(M).
- The injective hull E(M) is a maximal essential extension of M in the sense that if M⊆E(M) ⊊B for a module B, then M is not an essential submodule of B.
- The injective hull E(M) is a minimal injective module containing M in the sense that if M⊆B for an injective module B, then E(M) is (isomorphic to) a submodule of B.
- If N is an essential submodule of M, then E(N)=E(M).
- Every module M has an injective hull. The dual notion of a projective cover does not always exist for a module, however a flat cover exists for every module.
Read more about this topic: Injective Hull
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)