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Concept
Injective hull
Summary
In mathematics, particularly in algebra, the injective hull (or injective envelope) of a module is both the smallest injective module containing it and the largest essential extension of it. Injective hulls were first described in .
A module E is called the injective hull of a module M, if E is an essential extension of M, and E is injective. Here, the base ring is a ring with unity, though possibly non-commutative.
An injective module is its own injective hull.
The injective hull of an integral domain is its field of fractions .
The injective hull of a cyclic p-group (as Z-module) is a Prüfer group .
The injective hull of R/rad(R) is Homk(R,k), where R is a finite-dimensional k-algebra with Jacobson radical rad(R) .
A simple module is necessarily the socle of its injective hull.
The injective hull of the residue field of a discrete valuation ring where is .
In particular, the injective hull of in is the module .
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. A construction of the injective hull in terms of homomorphisms Hom(I, M), where I runs through the ideals of R, is given by .
The dual notion of a projective cover does not always exist for a module, however a flat cover exists for every module.
In some cases, for R a subring of a self-injective ring S, the injective hull of R will also have a ring structure.
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