In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, especially in the area of abstract algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

 known as module theory, the injective hull (or injective envelope) of a module is both the smallest injective module

Injective module

In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers...

 containing it and the largest essential extension of it. Injective hulls were first described in , and are described in detail in the textbook .

Definition

A module

Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

 E is called the injective hull of a module M, if E is an essential extension of M, and E is injective

Injective module

In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers...

. 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 and are both injective hulls. Then there is an isomorphism such that .

Examples

• The injective hull of an injective module is itself.
• The injective hull of an integral domain is its field of fractions
  Field of fractions
  In abstract algebra, the field of fractions or field of quotients of an integral domain is the smallest field in which it can be embedded. The elements of the field of fractions of the integral domain R have the form a/b with a and b in R and b ≠ 0...
  
  ,
• The injective hull of a cyclic p-group (as Z-module) is a Prüfer group
  Prüfer group
  In mathematics, specifically in group theory, the Prüfer p-group or the p-quasicyclic group or p∞-group, Z, for a prime number p is the unique p-group in which every element has p pth roots. The group is named after Heinz Prüfer...
  
  ,
• The injective hull of R/rad(R) is Hom_k(R,k), where R is a finite dimensional k-algebra
  Algebra (ring theory)
  In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R....
  
   with Jacobson radical
  Jacobson radical
  In mathematics, more specifically ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is an ideal which consists of those elements in R which annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition yields the same...
  
   rad(R), .
• A simple module
  Simple module
  In mathematics, specifically in ring theory, the simple modules over a ring R are the modules over R which have no non-zero proper submodules. Equivalently, a module M is simple if and only if every cyclic submodule generated by a non-zero element of M equals M...
  
   is necessarily the socle
  Socle (mathematics)
  -Socle of a group:In the context of group theory, the socle of a group G, denoted Soc, is the subgroup generated by the minimal non-trivial normal subgroups of G. The socle is a direct product of minimal normal subgroups...
  
   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

Indecomposable module

In abstract algebra, a module is indecomposable if it is non-zero and cannot be written as a direct sum of two non-zero submodules.Indecomposable is a weaker notion than simple module:simple means "no proper submodule" N...

.

Generalization

More generally, let C be an abelian category

Abelian category

In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...

. An object E is an injective hull of an object M if M → E is an essential extension and E is an injective object

Injective object

In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in homotopy theory and in theory of model categories...

. If C is locally small, satisfies Grothendieck's axiom AB5)

Abelian category

In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...

 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).

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