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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 R_/R for the nonzero prime ideals \mathfrak. 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 \hat R_P 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'' '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 R is a Noetherian ring and \mathfrak is a prime ideal, set E = E(R/\mathfrak) as the injective hull. The injective hull of R/\mathfrak over the Artinian ring R/\mathfrak^k can be computed as the module (0:_E\mathfrak^k). It is a module of the same length as R/\mathfrak^k. In particular, for the standard graded ring R_\bullet = k _1,\ldots,x_n\bullet and \mathfrak=(x_1,\ldots, x_n), E = \oplus_i \text(R_i, k) is an injective module, giving the tools for computing the indecomposable injective modules for artinian rings over k.


Self-injectivity

An Artin local ring (R, \mathfrak, K) is injective over itself if and only if soc(R) is a 1-dimensional vector space over K. 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 R = \mathbb ,y(x^2,xy,y^2) which has maximal ideal (x,y) and residue field \mathbb. Its socle is \mathbb\cdot x \oplus\mathbb\cdot y, which is 2-dimensional. The residue field has the injective hull \text_\mathbb(\mathbb\cdot x\oplus\mathbb\cdot y, \mathbb).


Modules over Lie algebras

For a Lie algebra \mathfrak over a field k of characteristic 0, the category of modules \mathcal(\mathfrak) has a relatively straightforward description of its injective modules. Using the universal enveloping algebra any injective \mathfrak-module can be constructed from the \mathfrak-module
\text_k(U(\mathfrak), V)
for some k-vector space V. Note this vector space has a \mathfrak-module structure from the injection
\mathfrak \hookrightarrow U(\mathfrak)
In fact, every \mathfrak-module has an injection into some \text_k(U(\mathfrak), V) and every injective \mathfrak-module is a direct summand of some \text_k(U(\mathfrak), V).


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 ...
R, 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 \mathfrak. That is, for an injective I \in \text(R) , there is an isomorphism
I \cong \bigoplus_ E(R/\mathfrak_i)
where E(R/\mathfrak_i) are the injective hulls of the modules R/\mathfrak_i. In addition, if I is the injective hull of some module M then the \mathfrak_i are the associated primes of M.


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''+ = HomZ(''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 f: S\to R 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 f_* M = \mathrm_S(R, M) 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

* * * * *


Primary sources

* * * * * * * * * * * {{DEFAULTSORT:Injective Module Homological algebra Module theory