A $C^{*}$-algebra $A$ has ideal property if any ideal $I$ of $A$ is generated as an ideal by the projections inside the ideal. Suppose that the limit $C^{*}$-algebra $A$ of inductive limit of direct sums of matrix algebras over spaces with uniformly bounded dimension has ideal property. In this paper we will prove that $A$ can be written as an inductive limit of certain very special subhomogeneous algebras, namely, direct sum of dimension drop interval algebras and matrix algebras over 2-dimensional spaces with torsion $H^{2}$ groups. This result unifies such reduction theorems for real rank zero $AH$ algebras in [EGS] and [DG] and for simple $AH$ algebras in [Li4]. The result play essential role in the classification of $AH$ algebra with ideal property (see [GJL]). 查看全文>>