Wikipedia / Injective hull
Wikipedia / Injective hull
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 .
Definition
A module is called the injective hull is an essential extension is injective. Here, the base ring is a ring with unity, though possibly non-commutative.
Examples
An injective module is its own injective hull.
The injective hull of an integral domain (as a module over itself) is its field of fractions .
-module) is a Prüfer group .
The injective hull of a torsion-free abelian group is the tensor product.
-algebra with Jacobson radical .
A simple module is necessarily the socle of its injective hull.
The injective hull of the residue field of a discrete valuation ring.
.
Properties
, however the isomorphism is not necessarily unique. This is because the injective hull's map extension property is not a full-fledged universal property.
) is a maximal essential extension.
.
.
, 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.
Ring structure
will also have a ring structure. to be a full matrix ring to be the ring of all upper triangular matrices. However, it is not always the case that the injective hull of a ring has a ring structure, as an example in shows.
A large class of rings which do have ring structures on their injective hulls are the nonsingular rings. In particular, for an integral domain, the injective hull of the ring (considered as a module over itself) is the field of fractions. The injective hulls of nonsingular rings provide an analogue of the ring of quotients for non-commutative rings, where the absence of the Ore condition may impede the formation of the classical ring of quotients. This type of "ring of quotients" (as these more general "fields of fractions" are called) was pioneered in , and the connection to injective hulls was recognized in .
Uniform dimension and injective modules
has finite uniform dimension (=finite rank indecomposable submodules.
Generalization
be an abelian category. An object is an injective hull is an injective object.
is locally small, satisfies Grothendieck's axiom AB5 and has enough injectives has an injective hull (these three conditions are satisfied by the category of modules over a ring). Every object in a Grothendieck category has an injective hull.
See also
Flat cover, the dual concept of injective hulls.
Rational hull: This is the analogue of the injective hull when considering a maximal rational extension.
Notes
References
Matsumura, H. Commutative Ring Theory, Cambridge studies in advanced mathematics volume 8.
External links
injective hull (PlanetMath article)