Injective hull
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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.
Properties
Every module M has an injective hull which is unique up to isomorphism. To be explicit, suppose [f_1:M \hookrightarrow E_1] and [f_2:M \hookrightarrow E_2] are both injective hulls. Then there is a unique isomorphism [\phi: E_1 \to E_2] such that [\phi\circ f_1 = f_2].
Examples
The injective hull of an injective module is itself.
The injective hull of an integral domain is its field of fractions.
External link
- [injective hull] (PlanetMath article)
Further reading
- Matsumura, H. Commutative Ring Theory, Cambridge studies in advanced mathematics volume 8.
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