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Reference.org
Algebraically compact module
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, algebraically compact modules, also called pure-injective modules, are <a href="/facts/Module_(mathematics)/jP0LIhFL">modules</a> 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 <a href="/facts/Module_homomorphism/SCSzvgxx">module homomorphisms</a>. These algebraically compact modules are analogous to <a href="/facts/Injective_module/1sAYFnyJ">injective modules</a>, 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. </p>