Injective hull
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In mathematics, especially in the area of abstract algebra known as module theory, 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), and are described in detail in the textbook (Lam 1999).
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[edit] Definition
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.
[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, (Lam 1999, Example 3.35)
- The injective hull of a cyclic p-group (as Z-module) is a Prüfer group, (Lam 1999, Example 3.36)
- The injective hull of R/rad(R) is Homk(R,k), where R is a finite dimensional k-algebra with Jacobson radical rad(R), (Lam 1999, Example 3.41).
- A simple module is necessarily the socle of its injective hull.
[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] References
- Eckmann, B.; Schopf, A. (1953), "Über injektive Moduln", Archiv der Mathematik 4: 75–78, doi:, MR0055978, ISSN 0003-9268
- Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, MR1653294, ISBN 978-0-387-98428-5
- Matsumura, H. Commutative Ring Theory, Cambridge studies in advanced mathematics volume 8.
