The purpose of this paper is to decipher constructively a lemma of Suslin which played a central role in his second solution of Serre’s problem on projective modules over polynomial rings. This lemma says that for a commutative ring
if
where
abcf1e90ab4617df4bff6"" title=""Click to view the MathML source"">v1 is monic and
n≥3, then there exist
such that, denoting by
wi the first coordinate of
, we have
. By the constructive proof we give, Suslin’s proof of Serre’s problem becomes fully constructive. Moreover, the new method with which we treat this academic example may be a model for miming constructively abstract proofs in which one works modulo a generic maximal ideal in order to prove that an ideal contains 1.