In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, especially in the area of abstract algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

 known as module theory, algebraically compact modules, also called pure-injective modules, are modules

Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

 that have a certain "nice" property which allows the solution of infinite systems of equations in the module by finitary means. The solutions to these systems allow the extension of certain kinds of module homomorphisms. These algebraically compact modules are analogous to injective module

Injective module

In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers...

s, where one can extend all module homomorphisms. All injective modules are algebraically compact, and the analogy between the two is made quite precise by a category embedding.

Definitions

Suppose R is a ring

Ring (mathematics)

In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

 and M is a left R-module. Take two sets I and J, and for every i in I and j in J, an element r_ij of R such that, for every i in I, only finitely many r_ij are non-zero. Furthermore, take an element m_i of M for every i in I. These data describe a system of linear equations in M:
  for every i∈I.

The goal is to decide whether this system has a solution, i.e. whether there exist elements x_j of M for every j in J such that all the equations of the system are simultaneously satisfied. (Note that we do not require that only finitely many of the x_j are non-zero here.)

Now consider such a system of linear equations, and assume that any subsystem consisting of only finitely many equations is solvable. (The solutions to the various subsystems may be different.) If every such "finitely-solvable" system is itself solvable, then we call the module M algebraically compact.

A module homomorphism M → K is called pure injective if the induced homomorphism between the tensor product

Tensor product

In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...

s C ⊗ M → C ⊗ K is injective for every right R-module C. The module M is pure-injective if any pure injective homomorphism j : M → K splits (i.e. there exists f : K → M with fj = 1_M).

It turns out that a module is algebraically compact if and only if it is pure-injective.

Examples

Every vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 is algebraically compact (since it is pure-injective). More generally, every injective module

Injective module

In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers...

 is algebraically compact, for the same reason.

If R is an associative algebra

Associative algebra

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...

 with 1 over some field

Field (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

 k, then every R-module with finite k-dimension is algebraically compact. This gives rise to the intuition that algebraically compact modules are those (possibly "large") modules which share the nice properties of "small" modules.

The Prüfer group

Prüfer group

In mathematics, specifically in group theory, the Prüfer p-group or the p-quasicyclic group or p∞-group, Z, for a prime number p is the unique p-group in which every element has p pth roots. The group is named after Heinz Prüfer...

s are algebraically compact abelian group

Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

s (i.e. Z-modules).

Many algebraically compact modules can be produced using the injective cogenerator

Injective cogenerator

In category theory, the concept of an injective cogenerator is drawn from examples such as Pontryagin duality. Generators are objects which cover other objects as an approximation, and cogenerators are objects which envelope other objects as an approximation...

 Q/Z of abelian groups. If H is a right module over the ring R, one forms the (algebraic) character module H* consisting of all group homomorphism

Group homomorphism

In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...

s from H to Q/Z. This is then a left R-module, and the *-operation yields a faithful contravariant functor

Functor

In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....

 from right R-modules to left R-modules.
Every module of the form H* is algebraically compact. Furthermore, there are pure injective homomorphisms H → H**, natural

Natural transformation

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...

 in H. One can often simplify a problem by first applying the *-functor, since algebraically compact modules are easier to deal with.

Facts

The following condition is equivalent to M being algebraically compact:

• For every index set I, the addition map M^(I) → M can be extended to a module homomorphism M^I → M (here M^(I) denotes the direct sum
  Direct sum of modules
  In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The result of the direct summation of modules is the "smallest general" module which contains the given modules as submodules...
  
   of copies of M, one for each element of I; M^I denotes the product
  Product (category theory)
  In category theory, the product of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces...
  
   of copies of M, one for each element of I).

Every indecomposable

Indecomposable module

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 is a weaker notion than simple module:simple means "no proper submodule" N...

 algebraically compact module has a local

Local ring

In abstract algebra, more particularly 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 varieties or manifolds, or of algebraic number fields examined at a particular place, or...

 endomorphism ring

Endomorphism ring

In abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object; this may be denoted End...

.

Algebraically compact modules share many other properties with injective objects because of the following: there exists an embedding of R-Mod into a Grothendieck category

Grothendieck category

In mathematics, a Grothendieck category is an AB5 category with a generator. In other words, it is an abelian category A admitting arbitrary coproducts, for which filtered colimits of exact sequences are exact and which possess a generator, i.e...

G under which the algebraically compact R-modules precisely correspond to the injective objects in G.