Injective hull
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This article is about the injective hull of a module in algebra. For injective hulls of metric spaces, also called tight spans, injective envelopes, or hyperconvex hulls, see tight span.
In mathematics, a module E is called the injective hull (or injective envelope) 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.
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[edit] Properties
Every module M has an injective hull which is unique up to isomorphism. To be explicit, suppose 
and 
are both injective hulls. Then there is an isomorphism 
such that 
.
[edit] Examples
The injective hull of an injective module is itself. The injective hull of an integral domain is its field of fractions.
[edit] Finite rank
The module M has finite rank if its injective hull is a finite direct sum of indecomposable submodules.
[edit] External links
- injective hull (PlanetMath article)
- PlanetMath page on modules of finite rank
[edit] Further reading
- Matsumura, H. Commutative Ring Theory, Cambridge studies in advanced mathematics volume 8.
