Absolutely pure modules

Source abstract

A module A A is shown to be absolutely pure if and only if every finite consistent system of linear equations over A A has a solution in A A . Noetherian, semihereditary, regular and Prüfer rings are characterized according to properties of absolutely pure modules over these rings. For example, R R is Noetherian if and only if every absolutely pure R R -module is injective and semihereditary if and only if the class of absolutely pure R R -modules is closed under homomorphic images. If R R is a Prüfer domain, then the absolutely pure R R -modules are the divisible modules and Ext R 1 ⁡ ( M , A ) = 0 \operatorname {Ext} _R^1(M,A) = 0 whenever A A is divisible and M M is a countably generated torsion-free R R -module.