Injective module - Citizendia

In mathematics, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and 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 In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions 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. 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 The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction 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 This concept is dual to that of projective modules. In Mathematics, particularly in Abstract algebra and Homological algebra, the concept of projective module over a ring R is a more flexible generalisation Injective modules were introduced by Reinhold Baer in 1940. 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.

1 Definition
2 Examples
3 Injective dimension
4 Facts
5 Generalization
6 References
7 See also

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:

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. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction 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:

commutative diagram defining injective module Q

Any short exact sequence 0 →Q → M → K → 0 of left R-modules splits. In mathematics and especially in Category theory a commutative diagram is a Diagram of objects also known as vertices, and Morphisms also In Mathematics, especially in Homological algebra and other applications of Abelian category theory as well as in Differential geometry and Group In Mathematics, especially in Homological algebra and other applications of Abelian category theory as well as in Differential geometry and Group
The contravariant functor Hom(-,Q) from the category of left R-modules to the category of abelian groups is exact. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Homological algebra, an exact functor is a Functor, from some category to another which preserves Exact sequences Exact functors are very

Injective right R-modules are defined in complete analogy.

Examples

Trivially, the zero module {0} is injective.

Given a field k, every k-vector space Q is an injective k-module. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Reason: if Q is a subspace of V, we can find a basis of Q and extend it to a basis of V. Basis vector redirects here For basis vector in the context of crystals see Crystal structure. The new extending basis vectors span a subspace K of V and V is the internal direct sum of Q and K. In the mathematical subfield of Linear algebra, the linear span, also called the linear hull, of a set of vectors in a Vector 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.

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. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of Translated into module language, this means that all modules over the group algebra kG are injective. In Mathematics, the group algebra is any of various constructions to assign to a Locally compact group an Operator algebra (or more generally a Banach 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 over k, then Hom_k(−, k) is a duality between finitely generated left A-modules and finitely generated right A-modules. In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive 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 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.

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

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 $\operatorname{Ext}^N_A(-,M) = 0$ for all N > n.

Facts

Any product of (even infinitely many) injective modules is injective. In Category theory, the product of two (or more objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as 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. The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction

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. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the e. an injective module over Z). More generally, an abelian group is injective if and only if it is divisible. In Mathematics, especially in the field of Group theory, a divisible group is an Abelian group in which every element can in some sense be divided by 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). In Abstract algebra, a principal ideal domain, or PID is an Integral domain in which every ideal is principal i Over a general integral domain, we still have one implication: every injective module over an integral domain is divisible.

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. 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 This is the prototype 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.

One can then go on to define the injective hull of a module (essentially the smallest injective module containing the given one). In Mathematics, especially in the area of Abstract algebra known as Module theory, the injective hull (or injective envelope) of a module is Every module M also has an injective resolution: an exact sequence of the form

0 → M → I⁰ → I¹ → I² → . In Mathematics, especially in Homological algebra and other applications of Abelian category theory as well as in Differential geometry and Group . .

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 indecomposable if it is non-zero and cannot be written as a Direct sum of two non-zero Submodules Indecomposable In Mathematics, more particularly in Abstract algebra, local rings are certain rings that are comparatively simple and serve to describe what is called In Abstract algebra, one associates to certain objects a ring, the object's 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 O_X-modules over some ringed space (X,O_X). In Mathematics, especially in the field of Category theory, the concept of injective object is a generalization of the concept of Injective module. In Category theory, a branch of Mathematics, the Functors between two given categories can themselves be turned into a category the morphisms in this functor In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. In Mathematics, a ringed space is intuitively speaking a space together with a collection of Commutative rings the elements of which are "functions" on 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 the context of Abstract algebra or Universal algebra, a monomorphism is simply an Injective Homomorphism.

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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