Projective modules

Abstract

In this note we prove that if R R is a ring satisfying a polynomial identity and P P is a projective left R R -module such that P P is finitely generated modulo the Jacobson radical, then P P is finitely generated. As a corollary we get that if R R is a ring still satisfying a polynomial identity and M M is a finitely generated flat R R -module such that M / J M M/JM is R / J R/J -projective, then M M is R R -projective, J J denotes the Jacobson radical.