Abstract
We want to define a deformation of the flag variety of SL
n
(k) the special linear group of k
n
, where kis an algebraically closed field of characteristic 0. We will construct the quantum flag ring as a subalgebra of k,
q
, [SL
n(k)], the quantum SL
n
(k) and we then will exhibit it in terms of generators and relations in the case in which k
q
is replaced by a suitable local ring in k(q). These results are a generalization of those obtained for the quantum grassmannian in [4].