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
* Modul ...
is both the smallest
injective module containing it and the largest
essential extension In mathematics, specifically module theory, given a ring ''R'' and an ''R''-module ''M'' with a submodule ''N'', the module ''M'' is said to be an essential extension of ''N'' (or ''N'' is said to be an essential submodule or large submodule of ''M' ...
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
* Modul ...
''E'' is called the injective hull of a module ''M'', if ''E'' is an
essential extension In mathematics, specifically module theory, given a ring ''R'' and an ''R''-module ''M'' with a submodule ''N'', the module ''M'' is said to be an essential extension of ''N'' (or ''N'' is said to be an essential submodule or large submodule of ''M' ...
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 is its
field of fractions .
* 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 rad(''R'') .
* A
simple module is necessarily the
socle of its injective hull.
* The injective hull of the residue field of a
discrete valuation ring 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 In mathematics, specifically module theory, given a ring ''R'' and an ''R''-module ''M'' with a submodule ''N'', the module ''M'' is said to be an essential extension of ''N'' (or ''N'' is said to be an essential submodule or large submodule of ''M' ...
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 In the branch of abstract mathematics called category theory, a projective cover of an object ''X'' is in a sense the best approximation of ''X'' by a projective object ''P''. Projective covers are the dual of injective envelopes.
Definition
L ...
does ''not'' always exist for a module, however a
flat cover 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 not ...
s. In particular, for an
integral domain, the injective hull of the ring (considered as a module over itself) is the
field of fractions. 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 may impede the formation of the
classical ring of quotients
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, o ...
. 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 (=''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
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 ai ...
''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 categories. ...
.
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 in ...
has an injective hull.
See also
*
Flat cover, 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