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In mathematics, 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 of some other module, then it is already a direct summand of that module; also, given a submodule of a module Y, then 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 by Reinhold Baer in 1940. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ...
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ...
In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ...
In abstract algebra, a module is a generalization of a vector space. ...
In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...
In abstract algebra, a module is a generalization of a vector space. ...
In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module (that is, a module with basis vectors). ...
Reinhold Baer (July 22, 1902 – October 22, 1979) was a German mathematician, known for his work in algebra. ...
Year 1940 (MCMXL) was a leap year starting on Monday (link will display the full 1940 calendar) of the Gregorian calendar. ...
Definition
More formally, a left module Q over the ring R is injective if it satisfies one (and therefore all) of the following equivalent conditions: In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...
- 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 = {0}.
- If X is a submodule of the left R-module Y and g : X → Q is a module homomorphism, then there exists a module homomorphism h : Y → Q such that h(x) = g(x) for all x in X.
- If X and Y are left-R modules and f : X → Y is an injective 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:
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 Injective right R-modules are defined in complete analogy. In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...
In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...
In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ...
Image File history File links No higher resolution available. ...
In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ...
In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ...
For functors in computer science, see the function object article. ...
In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...
In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ...
In homological algebra, an exact functor is one which preserves exact sequences. ...
Examples Trivially, the zero module {0} is injective.
Given a field k, every k-vector space 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 g in the above definition is typically not unique. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...
In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V...
In the mathematical subfield of linear algebra, the linear span, also called the linear hull, of a set of vectors in a vector space is the intersection of all subspaces containing that set. ...
If G is a finite group and k a field with characteristic 0, then one shows in the theory of group representations 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. This picture illustrates how the hours on a clock form a group under modular addition. ...
In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ...
Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ...
In the theory of group representations, the group algebra is any of various constructions to assign to a group (either a locally compact topological group, or a group without a topology, i. ...
If A is a unital associative algebra over the field k with finite dimension over k, then Homk(−, 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 Homk(P, k) where P is a finitely generated projective right A-module. In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ...
In mathematics, the dimension of a vector space V is the cardinality (i. ...
In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are essentially the same. There are numerous examples of categorical equivalences from many areas of mathematics. ...
Over other rings, injective modules are abundant, but it is not easy to come up with examples without some theory (mentioned below). The rationals Q (with addition) form an injective abelian group (i.e. an injective Z-module). The factor group Z/nZ for n > 1 is injective as a Z/nZ-module, but not injective as an abelian group. In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is a group that intuitively collapses the normal subgroup N to the identity element. ...
Injective dimension The injective dimension of an A-module M is the infimum of lengths of an injective resolution of M. It may take the value ∞. Equivalently, it is the minimal integer (if there is such, otherwise ∞) n such that  for all N > n.
Facts Any product of (even infinitely many) injective modules 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. In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...
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. In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ...
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. More generally still: a module over a principal ideal domain 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. In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ...
In group theory, a divisible group is an abelian group G such that for any positive integer n and any g in G, there exists y in G such that ny = g. ...
In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element). ...
Maybe the most important injective module is the abelian group Q/Z. It is an injective cogenerator in the category of abelian groups, 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 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. In category theory, the concept of an injective cogenerator is drawn from examples such as Pontryagin duality. ...
In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. ...
One can then go on to define the injective hull of a module (essentially the smallest injective module containing the given one). Every module M also has an injective resolution: an exact sequence of the form In mathematics, a module E is called the injective hull of a module M, if E is an essential extension of M, and E is injective. ...
In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ...
- 0 → M → I0 → I1 → I2 → ...
where the I j are injective. These injective resolutions are used to define the injective dimension of a module (the length of the shortest injective resolution ending in zeros, if such a finite resolution exists) as well as derived functors. In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. ...
Every indecomposable injective module has a local endomorphism ring. In abstract algebra, a module is defined to be indecomposable if it is non-zero and cannot be written as a direct sum of two non-zero submodules. ...
In mathematics, more particularly in abstract algebra, local rings are certain rings that are comparatively simple, and serve to describe the local behavior of functions defined on varieties or manifolds. ...
In abstract algebra, one associates to certain objects a ring, the objects endomorphism ring, which encodes several internal properties of the object. ...
Generalization One also talks about injective objects in categories more general than module categories, for instance in functor categories or in categories of sheaves of OX-modules over some ringed space (X,OX). The following general definition is used: an object Q of the category C is injective if for any monomorphism f : X → Y in C and any morphism g : X → Q there exists a morphism h : Y → Q with hf = g. In category theory, an object Q is said to be injective if every arrow to Q can be pushed forward across monomorphisms. ...
In category theory, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors. ...
In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and...
In mathematics, a ringed space is, intuitively speaking, a space together with, for each of its open sets, a commutative ring the elements of which are thought of as functions defined on that open set. ...
For other uses, see Dimorphism (disambiguation) or Polymorphism (disambiguation). ...
References - F.W. Anderson and K.R. Fuller: Rings and Categories of Modules, Graduate Texts in Mathematics, Vol. 13, 2nd Ed., Springer-Verlag, New York, 1992.
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