In mathematics, algebraically compact modules, also called pure-injective modules, are

modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...

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 homomorphism In algebra, a module homomorphism is a function between modules that preserves the module structures. Explicitly, if ''M'' and ''N'' are left modules over a ring ''R'', then a function f: M \to N is called an ''R''-''module homomorphism'' or an ' ...

s. These algebraically compact modules are analogous to

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. Specifically, if ''Q'' is a submodule ...

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 be a

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

, and a left -module. Consider a system of infinitely many linear equations :

\sum_ r_x_j = m_i,

where both sets and may be infinite,

m_i\in M,

and for each the number of nonzero

r_\in R

is finite. The goal is to decide whether such a system has a ''solution'', that is whether there exist elements of such that all the equations of the system are simultaneously satisfied. (It is not required that only finitely many 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

is a ''pure embedding'' if the induced homomorphism between the

tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...

s is

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; that is, implies . (Equivalently, implies in the equivalent contraposi ...

for every right -module . The module is pure-injective if any pure injective homomorphism

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(that is, there exists with

f\circ j=1_M

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

Examples

All modules with finitely many elements are algebraically compact. Every

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...

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

is algebraically compact, for the same reason. If ''R'' is an associative algebra with 1 over some

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''k'', then every ''R''-module with finite ''k''-

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 coor ...

is algebraically compact. This, together with that 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 In mathematics, specifically in group theory, the Prüfer ''p''-group or the ''p''-quasicyclic group or ''p''∞-group, Z(''p''∞), for a prime number ''p'' is the unique ''p''-group in which every element has ''p'' different ''p''-th roots. ...

s are algebraically compact

abelian group In mathematics, 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 the order in which they are written. That is, the group operation is com ...

s (i.e. Z-modules). The ring of ''p''-adic integers for each prime ''p'' is algebraically compact as both a module over itself and a module over Z. The

rational numbers 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 (e.g. ). The set of all r ...

are algebraically compact as a Z-module. Together with the

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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 In category theory, a branch of mathematics, 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 (dually) cogenerators are objects wh ...

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

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

functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, an ...

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''**,

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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 The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mo ...

of copies of ''M'', one for each element of ''I''; ''M^I'' denotes the

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of copies of ''M'', one for each element of ''I''). Every

algebraically compact module has a

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endomorphism ring In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in ...

. 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 In mathematics, a Grothendieck category is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957English translation in order to develop the machinery of homological algebra for modules and for sheaves i ...

''G'' under which the algebraically compact ''R''-modules precisely correspond to the injective objects in ''G''. Every ''R''-module is elementary equivalent to an algebraically compact ''R''-module and to a direct sum of

algebraically compact ''R''-modules.

References

{{reflist * C.U. Jensen and H. Lenzing: ''Model Theoretic Algebra'', Gordon and Breach, 1989 Module theory Model theory