Hereditary rings with countably generated cotorsion envelope

Abstract

Let R be a left hereditary ring. We show that if the left cotorsion envelope C(RR) of R is countably generated, then R is a semilocal ring. In particular, we deduce that C(RR) is finitely generated if and only if R is a semiperfect cotorsion ring. Our proof is based on set theoretical counting arguments. We also discuss some possible extensions of this result.