Let I be an ideal of a commutative ring A, B=A/I. Given n>=2, we characterize the vanishing of the André-Quillen homology modules H_{p}(A,B,W) for all B-module W and for all p, 2<=p<=n, in terms of some canonical morphisms. As a corollary, we obtain a new proof of a theorem of André. Finally, we construct an example of an ideal I of a commutative ring A such that H_{2}(A,B,W)=0 and H_{3}(A,B,W)=W for all B-module W
