  Based on the Basis theorem of Bruhat--Chevalley [C] and the formula for
multiplying Schubert classes obtained in [D\QTR{group}{u}] and programed in
[DZ$_{\QTR{group}{1}}$], we introduce a new method computing the Chow rings of
flag varieties (resp. the integral cohomology of homogeneous spaces). The
method and results of this paper have been extended in [DZ$_{3}$, DZ$_{4}$] to
obtain the integral cohomology rings of all complete flag manifolds, and to
construct the integral cohomologies of Lie groups in terms of Schubert classes.
