- Algebraically compact module
In

mathematics , especially in the area ofabstract algebra known asmodule theory ,**algebraically compact modules**, also called**pure-injective modules**, are modules 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 ofmodule homomorphism s. These algebraically compact modules are analogous toinjective 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**Suppose "R" is a ring 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"::$sum\_\{jin\; J\}\; r\_\{ij\}x\_j\; =\; m\_i$ 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 system, 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 s "C" ⊗ "M" → "C" ⊗ "K" isinjective 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 is algebraically compact (since it is pure-injective). More generally, everyinjective module is algebraically compact, for the same reason.If "R" is an

associative algebra with 1 over some field "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 s are algebraically compactabelian group s (i.e.**Z**-modules).Many algebraically compact modules can be produced using the

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 allgroup homomorphism s from "H" to**Q**/**Z**. This is then a left "R"-module, and the *-operation yields a faithful contravariantfunctor 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 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 thedirect sum of copies of "M", one for each element of "I"; "M^{I}" denotes the product of copies of "M", one for each element of "I").Every indecomposable algebraically compact module has a local

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

Table of mathematical symbols **References*** C.U. Jenzen and H. Lenzing: "Model Theoretic Algebra", Gordon and Breach, 1989

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