Projective modules

Abstract:

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