Injective hull

In mathematics, particularly in algebra, the injective hull 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 of a module, if is an essential extension of, and 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 is its field of fractions.
The injective hull of a cyclic -group is a Prüfer group.
The injective hull of a torsion-free abelian group is the tensor product.
The injective hull of is, where is a finite-dimensional -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 where is.
In particular, the injective hull of in is the module.

Properties

The injective hull of is unique up to isomorphisms which are the identity on, 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.
The injective hull is a maximal essential extension of in the sense that if for a module, then is not an essential submodule of.
The injective hull is a minimal injective module containing in the sense that if for an injective module, then is a submodule of.
If is an essential submodule of, then.
Every module has an injective hull. A construction of the injective hull in terms of homomorphisms, where runs through the ideals of, 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

In some cases, for a subring of a self-injective ring, the injective hull of will also have a ring structure. For instance, taking to be a full matrix ring over a field, and taking to be any ring containing every matrix which is zero in all but the last column, the injective hull of the right -module is. For instance, one can take 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 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" was pioneered in, and the connection to injective hulls was recognized in.

Uniform dimension and injective modules

An module has finite uniform dimension if and only if the injective hull of is a finite direct sum of indecomposable submodules.

Generalization

More generally, let be an abelian category. An object is an injective hull of an object if → is an essential extension and is an injective object.
If is locally small, satisfies Grothendieck's axiom AB5 and has enough injectives, then every object in has an injective hull. Every object in a Grothendieck category has an injective hull.