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 (Eckmann & Schopf 1953).

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.

In some cases, for R a subring of a self-injective ring S, the injective hull of R will also have a ring structure.[2] For instance, taking S to be a full matrix ring over a field, and taking R to be any ring containing every matrix which is zero in all but the last column, the injective hull of the right R-module R is S. For instance, one can take R 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 (Osofsky 1964) shows.

A large class of rings which do have ring structures on their injective hulls are the nonsingular rings.[3] 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 (Utumi 1956), and the connection to injective hulls was recognized in (Lambek 1963).

An R module M has finite uniform dimension (=finite rank) n if and only if the injective hull of M is a finite direct sum of n indecomposable submodules.

More generally, let C be an abelian category. An object E is an injective hull of an object M if ME is an essential extension and E is an injective object.

If C is locally small, satisfies Grothendieck's axiom AB5 and has enough injectives, then every object in C has an injective hull (these three conditions are satisfied by the category of modules over a ring).[4] Every object in a Grothendieck category has an injective hull.