On Projective Modules over Semi-hereditary Rings

This theorem, already known for finitely generated projective modules[l, I, Proposition 6.1], has been recently proved for arbitrary projective modules over commutative semi-hereditary rings by I. Kaplansky [2], who raised the problem of extending it to the noncommutative case. We recall two results due to Kaplansky: Any projective module (over an arbitrary ring) is a direct sum of countably generated modules [2, Theorem l]. If any direct summand A of a countably generated module M is such that each element of N is contained in a finitely generated direct summand, then M is a direct sum of finitely generated modules [2, Lemma l]. According to these results, it is sufficient to prove the following proposition : Each element of the module P is contained in a finitely generated direct summand of P. Let F = P ®Q be a free module and x be an arbitrary element of P. Let x=XiXi+ • • • +Xnx„ be a representation of the element x in some base for the free module F and let G denote the free submodule