Abstract
A moduleAAis shown to be absolutely pure if and only if every finite consistent system of linear equations overAAhas a solution inAA. Noetherian, semihereditary, regular and Prüfer rings are characterized according to properties of absolutely pure modules over these rings. For example,RRis Noetherian if and only if every absolutely pureRR-module is injective and semihereditary if and only if the class of absolutely pureRR-modules is closed under homomorphic images. IfRRis a Prüfer domain, then the absolutely pureRR-modules are the divisible modules andExtR1(M,A)=0\operatorname {Ext} _R^1(M,A) = 0wheneverAAis divisible andMMis a countably generated torsion-freeRR-module.
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