In mathematics, 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.
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Properties
Every module M has an injective hull which is unique up to isomorphism. To be explicit, suppose <math>f_1:M \hookrightarrow E_1</math> and <math>f_2:M \hookrightarrow E_2</math> are both injective hulls. Then there is a unique isomorphism <math>\phi: E_1 \to E_2</math> such that <math>\phi\circ f_1 = f_2</math>.
Examples
The injective hull of an injective module is itself. The injective hull of an integral domain is its field of fractions.
Finite rank
The module M has finite rank if its injective hull is a finite direct sum of indecomposable submodules.
External links
- injective hull (PlanetMath article)
- PlanetMath page on modules of finite rank
Further reading
- Matsumura, H. Commutative Ring Theory, Cambridge studies in advanced mathematics volume 8.
