Algebraically compact module

In mathematics, 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 of module homomorphisms. These algebraically compact modules are analogous to injective modules, 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.

In mathematics, 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 of module homomorphisms. These algebraically compact modules are analogous to injective modules, 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. \sum_{j\in J} r_{i,j}x_j = m_i, where both sets and may be infinite, m_i\in M, and for each the number of nonzero r_{i,j}\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 module homomorphism is a pure embedding if the induced homomorphism between the tensor products is injective for every right -module . The module is pure-injective if any pure injective homomorphism splits (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.

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