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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 ^//-projective, then M is Rprojective, J denotes the Jacobson radical. 0. Introduction. In this note R denotes an associative ring with an identity element, J denotes the Jacobson radical of R and all modules considered are unitary fi-modules.In [6] D. Lazard proved that a projective module P over a commutative ring R, such that P/JP is finitely generated, is finitely generated. Furthermore he remarked that is was unknown whether or not the commutativity of R was essential for the validity of the result. In this note we prove that the theorem holds in any P.L ring.1. Flat and projective modules. Let us consider the following properties of a ring R.(i) Any projective left fi-module P, such that P/JP is finitely generated, is finitely generated.(ii) Any finitely generated submodule of a projective module is contained in a maximal submodule.(iii) Any finitely generated flat left fi-module M, such that M/JM is R/Jprojective, is projective.First we prove that if (i) holds for all projective left fi-modules, then (iii) will also hold. This result might be well known, but I have not been able to find it in the literature. Proposition 1.1. If(i) holds in the ring R, then (iii) will also hold.Proof. Suppose Af is a finitely generated flat left fi-module, such that M/JM is fi/Z-projective, and M is not projective. We have an exact sequence

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