In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, especially in the area of
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
known as
module theory, an injective module is a
module ''Q'' that shares certain desirable properties with the Z-module Q of all
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example,
The set of all ...
s. Specifically, if ''Q'' is a
submodule of some other module, then it is already a
direct summand of that module; also, given a submodule of a module ''Y'', any
module homomorphism from this submodule to ''Q'' can be extended to a homomorphism from all of ''Y'' to ''Q''. This concept is
dual to that of
projective modules. Injective modules were introduced in and are discussed in some detail in the textbook .
Injective modules have been heavily studied, and a variety of additional notions are defined in terms of them:
Injective cogenerators are injective modules that faithfully represent the entire category of modules. Injective resolutions measure how far from injective a module is in terms of the
injective dimension and represent modules in the
derived category.
Injective hulls are maximal
essential extensions, and turn out to be minimal injective extensions. Over a
Noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals. If the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
, every injective module is uniquely a direct sum of
indecomposable modules, and their structure is well understood. An injective module over one ring may be not injective over another, but there are well-understood methods of changing rings which handle special cases. Rings which are themselves injective modules have a number of interesting properties and include rings such as
group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the gi ...
s of
finite group
In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...
s over
fields. Injective modules include
divisible groups and are generalized by the notion of
injective objects in
category theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
.
Definition
A left module ''Q'' over the
ring ''R'' is injective if it satisfies one (and therefore all) of the following equivalent conditions:
* If ''Q'' is a submodule of some other left ''R''-module ''M'', then there exists another submodule ''K'' of ''M'' such that ''M'' is the
internal direct sum of ''Q'' and ''K'', i.e. ''Q'' + ''K'' = ''M'' and ''Q'' ∩ ''K'' = .
* Any
short exact sequence
In mathematics, an exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Im ...
0 →''Q'' → ''M'' → ''K'' → 0 of left ''R''-modules
splits.
* If ''X'' and ''Y'' are left ''R''-modules, ''f'' : ''X'' → ''Y'' is an
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 of its codomain; that is, implies (equivalently by contraposition, impl ...
module homomorphism and ''g'' : ''X'' → ''Q'' is an arbitrary module homomorphism, then there exists a module homomorphism ''h'' : ''Y'' → ''Q'' such that ''hf'' = ''g'', i.e. such that the following diagram
commutes:
::
* The
contravariant Hom functor
In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between object (category theory), objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applicati ...
Hom(-,''Q'') from the
category of left ''R''-modules to the category of
abelian groups is
exact.
Injective right ''R''-modules are defined in complete analogy.
Examples
First examples
Trivially, the zero module is injective.
Given a
field ''k'', every ''k''-
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
''Q'' is an injective ''k''-module. Reason: if ''Q'' is a subspace of ''V'', we can find a
basis of ''Q'' and extend it to a basis of ''V''. The new extending basis vectors
span a subspace ''K'' of ''V'' and ''V'' is the internal direct sum of ''Q'' and ''K''. Note that the direct complement ''K'' of ''Q'' is not uniquely determined by ''Q'', and likewise the extending map ''h'' in the above definition is typically not unique.
The rationals Q (with addition) form an injective abelian group (i.e. an injective Z-module). The
factor group Q/Z and the
circle group are also injective Z-modules. The factor group Z/''n''Z for ''n'' > 1 is injective as a Z/''n''Z-module, but ''not'' injective as an abelian group.
Commutative examples
More generally, for any
integral domain
In mathematics, 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 setting for studying divisibilit ...
''R'' with field of fractions ''K'', the ''R''-module ''K'' is an injective ''R''-module, and indeed the smallest injective ''R''-module containing ''R''. For any
Dedekind domain, the
quotient module ''K''/''R'' is also injective, and its
indecomposable summands are the
localizations for the nonzero
prime ideals
. The
zero ideal is also prime and corresponds to the injective ''K''. In this way there is a 1-1 correspondence between prime ideals and indecomposable injective modules.
A particularly rich theory is available for
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...
noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals. If the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
s due to
Eben Matlis, . Every injective module is uniquely a direct sum of indecomposable injective modules, and the indecomposable injective modules are uniquely identified as the injective hulls of the quotients ''R''/''P'' where ''P'' varies over the
prime spectrum of the ring. The injective hull of ''R''/''P'' as an ''R''-module is canonically an ''R''
''P'' module, and is the ''R''
''P''-injective hull of ''R''/''P''. In other words, it suffices to consider
local ring
In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on algebraic varieties or manifolds, or of ...
s. The
endomorphism ring of the injective hull of ''R''/''P'' is the
completion of ''R'' at ''P''.
Two examples are the injective hull of the Z-module Z/''p''Z (the
Prüfer group), and the injective hull of the ''k''
'x''module ''k'' (the ring of inverse polynomials). The latter is easily described as ''k''
−1">'x'',''x''−1''xk''
'x'' This module has a basis consisting of "inverse monomials", that is ''x''
−''n'' for ''n'' = 0, 1, 2, …. Multiplication by scalars is as expected, and multiplication by ''x'' behaves normally except that ''x''·1 = 0. The endomorphism ring is simply the ring of
formal power series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial su ...
.
Artinian examples
If ''G'' is a
finite group
In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...
and ''k'' a field with
characteristic 0, then one shows in the theory of
group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used ...
s that any subrepresentation of a given one is already a direct summand of the given one. Translated into module language, this means that all modules over the
group algebra ''kG'' are injective. If the characteristic of ''k'' is not zero, the following example may help.
If ''A'' is a unital
associative algebra over the field ''k'' with finite
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
over ''k'', then Hom
''k''(−, ''k'') is a
duality between finitely generated left ''A''-modules and finitely generated right ''A''-modules. Therefore, the finitely generated injective left ''A''-modules are precisely the modules of the form Hom
''k''(''P'', ''k'') where ''P'' is a finitely generated projective right ''A''-module. For
symmetric algebras, the duality is particularly well-behaved and projective modules and injective modules coincide.
For any
Artinian ring
In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are ...
, just as for
commutative ring
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
s, there is a 1-1 correspondence between prime ideals and indecomposable injective modules. The correspondence in this case is perhaps even simpler: a prime ideal is an annihilator of a unique simple module, and the corresponding indecomposable injective module is its
injective hull. For finite-dimensional algebras over fields, these injective hulls are
finitely-generated modules .
Computing injective hulls
If
is a Noetherian ring and
is a prime ideal, set
as the injective hull. The injective hull of
over the Artinian ring
can be computed as the module
. It is a module of the same length as
.
In particular, for the standard graded ring
and
,
is an injective module, giving the tools for computing the indecomposable injective modules for artinian rings over
.
Self-injectivity
An Artin local ring
is injective over itself if and only if
is a 1-dimensional vector space over
. This implies every local Gorenstein ring which is also Artin is injective over itself since has a 1-dimensional socle. A simple non-example is the ring
which has maximal ideal
and residue field
. Its socle is
, which is 2-dimensional. The residue field has the injective hull
.
Modules over Lie algebras
For a Lie algebra
over a field
of characteristic 0, the category of modules
has a relatively straightforward description of its injective modules. Using the universal enveloping algebra any injective
-module can be constructed from the
-module
for some
-vector space
. Note this vector space has a
-module structure from the injection
In fact, every
-module has an injection into some
and every injective
-module is a direct summand of some
.
Theory
Structure theorem for commutative Noetherian rings
Over a commutative
Noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals. If the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
, every injective module is a direct sum of indecomposable injective modules and every indecomposable injective module is the injective hull of the residue field at a prime
. That is, for an injective
, there is an isomorphism
where
are the injective hulls of the modules
. In addition, if
is the injective hull of some module
then the
are the associated primes of
.
Submodules, quotients, products, and sums, Bass-Papp Theorem
Any
product of (even infinitely many) injective modules is injective; conversely, if a direct product of modules is injective, then each module is injective . Every direct sum of finitely many injective modules is injective. In general, submodules, factor modules, or infinite
direct sums of injective modules need not be injective. Every submodule of every injective module is injective if and only if the ring is
Artinian semisimple ; every factor module of every injective module is injective if and only if the ring is
hereditary, .
Bass-Papp Theorem states that every infinite direct sum of right (left) injective modules is injective if and only if the ring is right (left)
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite leng ...
, .
Baer's criterion
In Baer's original paper, he proved a useful result, usually known as Baer's Criterion, for checking whether a module is injective: a left ''R''-module ''Q'' is injective if and only if any homomorphism ''g'' : ''I'' → ''Q'' defined on a
left ideal ''I'' of ''R'' can be extended to all of ''R''.
Using this criterion, one can show that Q is an injective
abelian group (i.e. an injective module over Z). More generally, an abelian group is injective if and only if it is
divisible
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...
. More generally still: a module over a
principal ideal domain
In mathematics, a principal ideal domain, or PID, is an integral domain (that is, a non-zero commutative ring without nonzero zero divisors) in which every ideal is principal (that is, is formed by the multiples of a single element). Some author ...
is injective if and only if it is divisible (the case of vector spaces is an example of this theorem, as every field is a principal ideal domain and every vector space is divisible). Over a general integral domain, we still have one implication: every injective module over an integral domain is divisible.
Baer's criterion has been refined in many ways , including a result of and that for a commutative Noetherian ring, it suffices to consider only
prime ideals ''I''. The dual of Baer's criterion, which would give a test for projectivity, is false in general. For instance, the Z-module Q satisfies the dual of Baer's criterion but is not projective.
Injective cogenerators
Maybe the most important injective module is the abelian group Q/Z. It is an
injective cogenerator in the
category of abelian groups
In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab.
Properties
The zero object o ...
, which means that it is injective and any other module is contained in a suitably large product of copies of Q/Z. So in particular, every abelian group is a subgroup of an injective one. It is quite significant that this is also true over any ring: every module is a submodule of an injective one, or "the category of left ''R''-modules has enough injectives." To prove this, one uses the peculiar properties of the abelian group Q/Z to construct an injective cogenerator in the category of left ''R''-modules.
For a left ''R''-module ''M'', the so-called "character module" ''M''
+ = Hom
Z(''M'',Q/Z) is a right ''R''-module that exhibits an interesting duality, not between injective modules and
projective modules, but between injective modules and
flat module
In algebra, flat modules include free modules, projective modules, and, over a principal ideal domain, torsion-free modules. Formally, a module (mathematics), module ''M'' over a ring (mathematics), ring ''R'' is ''flat'' if taking the tensor prod ...
s . For any ring ''R'', a left ''R''-module is flat if and only if its character module is injective. If ''R'' is left noetherian, then a left ''R''-module is injective if and only if its character module is flat.
Injective hulls
The
injective hull of a module is the smallest injective module containing the given one and was described in .
One can use injective hulls to define a minimal injective resolution (see below). If each term of the injective resolution is the injective hull of the cokernel of the previous map, then the injective resolution has minimal length.
Injective resolutions
Every module ''M'' also has an injective
resolution: an
exact sequence of the form
:0 → ''M'' → ''I''
0 → ''I''
1 → ''I''
2 → ...
where the ''I''
''j'' are injective modules. Injective resolutions can be used to define
derived functor
In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.
Motivation
It was noted in vari ...
s such as the
Ext functor.
The ''length'' of a finite injective resolution is the first index ''n'' such that ''I''
''n'' is nonzero and ''I''
''i'' = 0 for ''i'' greater than ''n''. If a module ''M'' admits a finite injective resolution, the minimal length among all finite injective resolutions of ''M'' is called its injective dimension and denoted id(''M''). If ''M'' does not admit a finite injective resolution, then by convention the injective dimension is said to be infinite. As an example, consider a module ''M'' such that id(''M'') = 0. In this situation, the exactness of the sequence 0 → ''M'' → ''I''
0 → 0 indicates that the arrow in the center is an isomorphism, and hence ''M'' itself is injective.
[A module isomorphic to an injective module is of course injective.]
Equivalently, the injective dimension of ''M'' is the minimal integer (if there is such, otherwise ∞) ''n'' such that Ext(–,''M'') = 0 for all ''N'' > ''n''.
Indecomposables
Every injective submodule of an injective module is a direct summand, so it is important to understand
indecomposable injective modules, .
Every indecomposable injective module has a
local
Local may refer to:
Geography and transportation
* Local (train), a train serving local traffic demand
* Local, Missouri, a community in the United States
Arts, entertainment, and media
* ''Local'' (comics), a limited series comic book by Bria ...
endomorphism ring. A module is called a ''
uniform module'' if every two nonzero submodules have nonzero intersection. For an injective module ''M'' the following are equivalent:
* ''M'' is indecomposable
* ''M'' is nonzero and is the injective hull of every nonzero submodule
* ''M'' is uniform
* ''M'' is the injective hull of a uniform module
* ''M'' is the injective hull of a uniform
cyclic module In mathematics, more specifically in ring theory, a cyclic module or monogenous module is a module over a ring that is generated by one element. The concept is a generalization of the notion of a cyclic group, that is, an Abelian group (i.e. Z ...
* ''M'' has a local endomorphism ring
Over a Noetherian ring, every injective module is the direct sum of (uniquely determined) indecomposable injective modules. Over a commutative Noetherian ring, this gives a particularly nice understanding of all injective modules, described in . The indecomposable injective modules are the injective hulls of the modules ''R''/''p'' for ''p'' a prime ideal of the ring ''R''. Moreover, the injective hull ''M'' of ''R''/''p'' has an increasing filtration by modules ''M''
''n'' given by the annihilators of the ideals ''p''
''n'', and ''M''
''n''+1/''M''
''n'' is isomorphic as finite-dimensional vector space over the quotient field ''k''(''p'') of ''R''/''p'' to Hom
''R''/''p''(''p''
''n''/''p''
''n''+1, ''k''(''p'')).
Change of rings
It is important to be able to consider modules over
subrings or
quotient ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
s, especially for instance
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...
s. In general, this is difficult, but a number of results are known, .
Let ''S'' and ''R'' be rings, and ''P'' be a left-''R'', right-''S''
bimodule
In abstract algebra, a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. Besides appearing naturally in many parts of mathematics, bimodules play a clarifying role, i ...
that is
flat as a left-''R'' module. For any injective right ''S''-module ''M'', the set of
module homomorphisms Hom
''S''( ''P'', ''M'' ) is an injective right ''R''-module. The same statement holds of course after interchanging left- and right- attributes.
For instance, if ''R'' is a subring of ''S'' such that ''S'' is a flat ''R''-module, then every injective ''S''-module is an injective ''R''-module. In particular, if ''R'' is an integral domain and ''S'' 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 fie ...
, then every vector space over ''S'' is an injective ''R''-module. Similarly, every injective ''R''
'x''module is an injective ''R''-module.
In the opposite direction, a ring homomorphism
makes ''R'' into a left-''R'', right-''S'' bimodule, by left and right multiplication. Being
free over itself ''R'' is also
flat as a left ''R''-module. Specializing the above statement for ''P = R'', it says that when ''M'' is an injective right ''S''-module the
coinduced module is an injective right ''R''-module. Thus, coinduction over ''f'' produces injective ''R''-modules from injective ''S''-modules.
For quotient rings ''R''/''I'', the change of rings is also very clear. An ''R''-module is an ''R''/''I''-module precisely when it is annihilated by ''I''. The submodule ann
''I''(''M'') = is a left submodule of the left ''R''-module ''M'', and is the largest submodule of ''M'' that is an ''R''/''I''-module. If ''M'' is an injective left ''R''-module, then ann
''I''(''M'') is an injective left ''R''/''I''-module. Applying this to ''R''=Z, ''I''=''n''Z and ''M''=Q/Z, one gets the familiar fact that Z/''n''Z is injective as a module over itself. While it is easy to convert injective ''R''-modules into injective ''R''/''I''-modules, this process does not convert injective ''R''-resolutions into injective ''R''/''I''-resolutions, and the homology of the resulting complex is one of the early and fundamental areas of study of relative homological algebra.
The textbook has an erroneous proof that
localization preserves injectives, but a counterexample was given in .
Self-injective rings
Every ring with unity is a
free module
In mathematics, a free module is a module that has a ''basis'', that is, a generating set that is linearly independent. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commu ...
and hence is a
projective as a module over itself, but it is rarer for a ring to be injective as a module over itself, . If a ring is injective over itself as a right module, then it is called a right self-injective ring. Every
Frobenius algebra is self-injective, but no
integral domain
In mathematics, 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 setting for studying divisibilit ...
that is not a
field is self-injective. Every proper
quotient
In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...
of a
Dedekind domain is self-injective.
A right
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite leng ...
, right self-injective ring is called a
quasi-Frobenius ring, and is two-sided
Artinian and two-sided injective, . An important module theoretic property of quasi-Frobenius rings is that the projective modules are exactly the injective modules.
Generalizations and specializations
Injective objects
One also talks about
injective objects in
categories more general than module categories, for instance in
functor categories or in categories of
sheaves of O
''X''-modules over some
ringed space
In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf of ...
(''X'',O
''X''). The following general definition is used: an object ''Q'' of the category ''C'' is injective if for any
monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y.
In the more general setting of category theory, a monomorphis ...
''f'' : ''X'' → ''Y'' in ''C'' and any morphism ''g'' : ''X'' → ''Q'' there exists a morphism ''h'' : ''Y'' → ''Q'' with ''hf'' = ''g''.
Divisible groups
The notion of injective object in the category of abelian groups was studied somewhat independently of injective modules under the term
divisible group. Here a Z-module ''M'' is injective if and only if ''n''⋅''M'' = ''M'' for every nonzero integer ''n''. Here the relationships between
flat module
In algebra, flat modules include free modules, projective modules, and, over a principal ideal domain, torsion-free modules. Formally, a module (mathematics), module ''M'' over a ring (mathematics), ring ''R'' is ''flat'' if taking the tensor prod ...
s,
pure submodules, and injective modules is more clear, as it simply refers to certain divisibility properties of module elements by integers.
Pure injectives
In relative homological algebra, the extension property of homomorphisms may be required only for certain submodules, rather than for all. For instance, a
pure injective module
In mathematics, algebraically compact modules, also called pure-injective modules, are module (mathematics), modules that have a certain "nice" property which allows the solution of infinite systems of equations in the module by finitary means. Th ...
is a module in which a homomorphism from a
pure submodule can be extended to the whole module.
References
Notes
Textbooks
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Primary sources
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{{DEFAULTSORT:Injective Module
Homological algebra
Module theory