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Absolutely pure modules
A module<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"><mml:semantics><mml:mi>A</mml:mi><mml:annotation encoding="application/x-tex">A</mml:annotation></mml:semantics></mml:math></inline-formula>is shown to be absolutely pure if and only if every finite consistent system of linear equations over<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"><mml:semantics><mml:mi>A</mml:mi><mml:annotation encoding="application/x-tex">A</mml:annotation></mml:semantics></mml:math></inline-formula>has a solution in<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"><mml:semantics><mml:mi>A</mml:mi><mml:annotation encoding="application/x-tex">A</mml:annotation></mml:semantics></mml:math></inline-formula>. Noetherian, semihereditary, regular and Prüfer rings are characterized according to properties of absolutely …