Injective hull

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, 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).

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.

Properties

Every module M has an injective hull which is unique up to isomorphism. To be explicit, suppose $f_1 \colon M \hookrightarrow E_1$ and $f_2 \colon M \hookrightarrow E_2$ are both injective hulls. Then there is an isomorphism $\phi \colon E_1 \to E_2$ 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, (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 Hom_k(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.

Uniform dimension and injective modules

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.

Generalization

More generally, let C be an abelian category. An object E is an injective hull of an object M if M → E 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).^[1]

See also

• Rational hull: This is the analogue of the injective hull when considering a maximal rational extension.

External links

• injective hull (PlanetMath article)
• PlanetMath page on modules of finite rank

Notes

1. ^ Section III.2 of (Mitchell 1965)

References

• Eckmann, B.; Schopf, A. (1953), "Über injektive Moduln", Archiv der Mathematik 4 (2): 75–78, doi:10.1007/BF01899665, ISSN 0003-9268, MR0055978 
• Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98428-5, MR1653294 
• Matsumura, H. Commutative Ring Theory, Cambridge studies in advanced mathematics volume 8.
• Mitchell, Barry (1965), Theory of categories, Pure and applied mathematics, 17, Academic Press, ISBN 978-0-124-99250-4, MR0202787