In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, particularly in
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
, the injective hull (or injective envelope) of a
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Mo ...
is both the smallest
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. Specifically, if ''Q'' is a submodule o ...
containing it and the largest
essential extension of it. Injective hulls were first described in .
Definition
A
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Mo ...
''E'' is called the injective hull of a module ''M'', if ''E'' is an
essential extension of ''M'', and ''E'' is
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
. Here, the base ring is a ring with unity, though possibly non-commutative.
Examples
* An injective module is its own injective hull.
* The injective hull of an
integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural set ...
is its
field of fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
.
* The injective hull of a cyclic ''p''-group (as Z-module) is a
Prüfer group
In mathematics, specifically in group theory, the Prüfer ''p''-group or the ''p''-quasicyclic group or ''p''∞-group, Z(''p''∞), for a prime number ''p'' is the unique ''p''-group in which every element has ''p'' different ''p''-th roots.
...
.
* The injective hull of ''R''/rad(''R'') is Hom
''k''(''R'',''k''), where ''R'' is a finite-dimensional ''k''-
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
with
Jacobson radical In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition yie ...
rad(''R'') .
* A
simple module In mathematics, specifically in ring theory, the simple modules over a ring ''R'' are the (left or right) modules over ''R'' that are non-zero and have no non-zero proper submodules. Equivalently, a module ''M'' is simple if and only if every cycl ...
is necessarily the
socle of its injective hull.
* The injective hull of the residue field of a
discrete valuation ring
In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.
This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions:
# ''R'' ...
where
is
.
* In particular, the injective hull of
in
is the module
.
Properties
* The injective hull of ''M'' is unique up to isomorphisms which are the identity on ''M'', however the isomorphism is not necessarily unique. This is because the injective hull's map extension property is not a full-fledged
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
. Because of this uniqueness, the hull can be denoted as ''E''(''M'').
* The injective hull ''E''(''M'') is a maximal
essential extension of ''M'' in the sense that if ''M''⊆''E''(''M'') ⊊''B'' for a module ''B'', then ''M'' is not an essential submodule of ''B''.
* The injective hull ''E''(''M'') is a minimal injective module containing ''M'' in the sense that if ''M''⊆''B'' for an injective module ''B'', then ''E''(''M'') is (isomorphic to) a submodule of ''B''.
* If ''N'' is an essential submodule of ''M'', then ''E''(''N'')=''E''(''M'').
* Every module ''M'' has an injective hull. A construction of the injective hull in terms of homomorphisms Hom(''I'', ''M''), where ''I'' runs through the ideals of ''R'', is given by .
* The dual notion of a
projective cover does ''not'' always exist for a module, however a
flat cover In algebra, a flat cover of a module ''M'' over a ring is a surjective homomorphism from a flat module ''F'' to ''M'' that is in some sense minimal. Any module over a ring has a flat cover that is unique up to (non-unique) isomorphism. Flat covers ...
exists for every module.
Ring structure
In some cases, for ''R'' a subring of a self-injective ring ''S'', the injective hull of ''R'' will also have a ring structure. For instance, taking ''S'' to be a full
matrix ring
In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ''U ...
over a field, and taking ''R'' to be any ring containing every matrix which is zero in all but the last column, the injective hull of the right ''R''-module ''R'' is ''S''. For instance, one can take ''R'' to be the ring of all upper triangular matrices. However, it is not always the case that the injective hull of a ring has a ring structure, as an example in shows.
A large class of rings which do have ring structures on their injective hulls are the
nonsingular ring In the branches of abstract algebra known as ring theory and module theory, each right (resp. left) ''R''-module ''M'' has a singular submodule consisting of elements whose annihilators are essential right (resp. left) ideals in ''R''. In set no ...
s. In particular, for an
integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural set ...
, the injective hull of the ring (considered as a module over itself) is the
field of fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
. The injective hulls of nonsingular rings provide an analogue of the ring of quotients for non-commutative rings, where the absence of the
Ore condition
In mathematics, especially in the area of algebra known as ring theory, the Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, or ...
may impede the formation of the
classical ring of quotients. This type of "ring of quotients" (as these more general "fields of fractions" are called) was pioneered in , and the connection to injective hulls was recognized in .
Uniform dimension and injective modules
An ''R'' module ''M'' has finite
uniform dimension In abstract algebra, a module is called a uniform module if the intersection of any two nonzero submodules is nonzero. This is equivalent to saying that every nonzero submodule of ''M'' is an essential submodule. A ring may be called a right (left ...
(=''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
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 prototypical example of an abelian category is the category of ab ...
. An
object
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ...
''E'' is an injective hull of an object ''M'' if ''M'' → ''E'' is an essential extension and ''E'' is an
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 cohomology, in homotopy theory and in the theory of model categori ...
.
If C is
locally small
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows asso ...
, satisfies
Grothendieck's axiom AB5 and has
enough injectives
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 cohomology, in homotopy theory and in the theory of model categori ...
, then every object in C has an injective hull (these three conditions are satisfied by the category of modules over a ring).
[Section III.2 of ] Every object in a
Grothendieck category In mathematics, a Grothendieck category is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957English translation in order to develop the machinery of homological algebra for modules and for sheaves ...
has an injective hull.
See also
*
Flat cover In algebra, a flat cover of a module ''M'' over a ring is a surjective homomorphism from a flat module ''F'' to ''M'' that is in some sense minimal. Any module over a ring has a flat cover that is unique up to (non-unique) isomorphism. Flat covers ...
, the dual concept of injective hulls.
*
Rational hull In abstract algebra, specifically in module theory, a dense submodule of a module is a refinement of the notion of an essential submodule. If ''N'' is a dense submodule of ''M'', it may alternatively be said that "''N'' ⊆ ''M'' is a rat ...
: This is the analogue of the injective hull when considering a maximal
rational extension.
Notes
References
*
*
*
*
*
* Matsumura, H. ''Commutative Ring Theory'', Cambridge studies in advanced mathematics volume 8.
*
*
*{{Citation , last1=Utumi , first1=Yuzo , title=On quotient rings , mr=0078966 , year=1956 , journal=Osaka Journal of Mathematics , issn=0030-6126 , volume=8 , pages=1–18
External links
injective hull(PlanetMath article)
PlanetMath page on modules of finite rank Module theory