Injective Hull - Properties

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.

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