# Algebraically compact module

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{{Short description|Pure-injective modules in mathematics}}
In [mathematics](/source/mathematics), '''algebraically compact modules''', also called '''pure-injective modules''', are [modules](/source/module_(mathematics)) that have a certain "nice" property which allows the solution of infinite systems of equations in the module by [finitary](/source/finitary) means.  The solutions to these systems allow the extension of certain kinds of [module homomorphism](/source/module_homomorphism)s.  These algebraically compact modules are analogous to [injective module](/source/injective_module)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 ==

Let {{math|''R''}} be a [ring](/source/ring_(mathematics)), and {{math|''M''}} a left {{math|''R''}}-module. Consider a system of infinitely many linear equations
:<math>\sum_{j\in J} r_{i,j}x_j = m_i,</math>
where both sets {{mvar|I}} and {{mvar|J}} may be infinite, <math>m_i\in M,</math> and for each {{mvar|i}} the number of nonzero <math>r_{i,j}\in R</math> is finite.

The goal is to decide whether such a system has a ''solution'', that is whether there exist elements {{math|''x''<sub>''j''</sub>}} of {{mvar|''M''}} such that all the equations of the system are simultaneously satisfied. (It is not required that only finitely many {{math|''x<sub>j</sub>''}} are non-zero.)

The module ''M'' is '''algebraically compact''' if, for all such systems, if every subsystem formed by a finite number of the equations has a solution, then the whole system has a solution. (The solutions to the various subsystems may be different.)

On the other hand, a [module homomorphism](/source/module_homomorphism) {{math|''M'' → ''K''}} is a ''pure embedding'' if the [induced homomorphism](/source/induced_homomorphism) between the [tensor product](/source/tensor_product)s {{math|''C'' ⊗ ''M'' → ''C'' ⊗ ''K''}}{{math|}} is [injective](/source/injective) for every right {{math|''R''}}-module {{math|''C''}}. The module {{math|''M''}} is '''pure-injective''' if any pure injective homomorphism {{math|''j'' : ''M'' → ''K''}} [splits](/source/split_short_exact_sequence) (that is, there exists {{math|''f'' : ''K'' → ''M''}} with <math>f\circ j=1_M</math>).

It turns out that a module is algebraically compact [if and only if](/source/if_and_only_if) it is pure-injective.

== Examples ==

All modules with finitely many elements are algebraically compact.

Every [vector space](/source/vector_space) is algebraically compact (since it is pure-injective). More generally, every [injective module](/source/injective_module) is algebraically compact, for the same reason.

If ''R'' is an [associative algebra](/source/associative_algebra) with 1 over some [field](/source/field_(mathematics)) ''k'', then every ''R''-module with finite ''k''-[dimension](/source/dimension_of_a_vector_space) is algebraically compact. This, together with the fact that all finite modules are algebraically compact, 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](/source/Pr%C3%BCfer_group)s are algebraically compact [abelian group](/source/abelian_group)s (i.e. '''Z'''-modules). The ring of [''p''-adic integers](/source/p-adic_number) for each prime ''p'' is algebraically compact as both a module over itself and a module over '''Z'''. The [rational numbers](/source/rational_number) are algebraically compact as a '''Z'''-module. Together with the [indecomposable](/source/indecomposable_module) finite modules over '''Z''', this is a complete list of indecomposable algebraically compact modules.

Many algebraically compact modules can be produced using the [injective cogenerator](/source/injective_cogenerator) '''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](/source/group_homomorphism)s from ''H'' to '''Q'''/'''Z'''. This is then a left ''R''-module, and the *-operation yields a [faithful](/source/faithful_functor) contravariant [functor](/source/functor) 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](/source/natural_transformation) 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<sup>(I)</sup>'' → ''M'' can be extended to a module homomorphism ''M<sup>I</sup>'' → ''M'' (here ''M<sup>(I)</sup>'' denotes the [direct sum](/source/direct_sum_of_modules) of copies of ''M'', one for each element of ''I''; ''M<sup>I</sup>'' denotes the [product](/source/product_(category_theory)) of copies of ''M'', one for each element of ''I'').

Every [indecomposable](/source/indecomposable_module) algebraically compact module has a [local](/source/local_ring) [endomorphism ring](/source/endomorphism_ring).

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](/source/Grothendieck_category) ''G'' under which the algebraically compact ''R''-modules precisely correspond to the injective objects in ''G''.

Every ''R''-module is [elementary equivalent](/source/elementary_equivalence) to an algebraically compact ''R''-module and to a direct sum of [indecomposable](/source/indecomposable_module) algebraically compact ''R''-modules.<ref>{{cite book|last1=Prest|first1=Mike|title=Model theory and modules|date=1988|publisher=Cambridge University Press, Cambridge|location=London Mathematical Society Lecture Note Series|isbn=0-521-34833-1}}</ref>

== References ==
{{reflist}}
* C.U. Jensen and H. Lenzing: ''Model Theoretic Algebra'', Gordon and Breach, 1989

<!--- Categories --->
Category:Module theory
Category:Model theory

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