A module A is shown to be absolutely pure if and only if every finite consistent system of linear equations over A has a solution in A. Noetherian, semihereditary, regular and Prüfer rings are characterized according to properties of absolutely pure modules over these rings. For example, R is Noetherian if and only if every absolutely pure i?-module is injective and semihereditary if and only if the class of absolutely pure i?-modules is closed under homomorphic images. If R is a Prüfer domain, then the absolutely pure i?-modules are the divisible modules and Ext\t(M, A)=0 whenever A is divisible and M is a countably generated torsionfree .R-module.

Abstract.A module A is shown to be absolutely pure if and only if every finite consistent system of linear equations over A has a solution in A. Noetherian, semihereditary, regular and Prüfer rings are characterized according to properties of absolutely pure modules over these rings. For example, R is Noetherian if and only if every absolutely pure i?-module is injective and semihereditary if and only if the class of absolutely pure i?-modules is closed under homomorphic images. If R is a Prüfer domain, then the absolutely pure i?-modules are the divisible modules and Ext\t(M, A)=0 whenever A is divisible and M is a countably generated torsionfree .R-module.

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