A Reduction theorem for $AH$ algebras with ideal property

Let $A$ be an $AH$ algebra, that is, $A$ is the inductive limit $C^{\ast }$-algebra of $$A_1\stackrel{{\varphi }_{1,2}}{\to }{A}_2\stackrel{{\varphi }_{2,3}}{\to }{A}_3⟶\cdots ⟶A_n⟶\cdots$$ with $A_n={⨁}_{i=1}^{t_n}{P}_{n,i}{M}_{[n,i]}(C(X_{n,i}))P_{n,i}$, where $X_{n,i}$ are compact metric spaces, $t_n$ and $[n,i]$ are positive integers, and $P_{n,i}\in M_{[n,i]}(C(X_{n,i}))$ are projections. Suppose that $A$ has the ideal property: each closed two-sided ideal of $A$ is generated by the projections inside the ideal, as a closed two-sided ideal. Suppose that ${\sup }_{n,i}dim(X_{n,i})<+\mathrm{\infty }$. In this article, we prove that $A$ can be written as the inductive limit of $$B_1⟶B_2⟶\cdots ⟶B_n⟶\cdots ,$$ where $B_n={⨁}_{i=1}^{s_n}{Q}_{n,i}{M}_{\{n,i\}}(C(Y_{n,i}))Q_{n,i}$, where $Y_{n,i}$ are $\{pt\}$, $[0,1]$,$S^1$,$T_{II,k},$ $T_{III,k}$ and $S^2$ (all of them are connected simplicial complexes of dimension at most three), $s_n$ and $\{n,i\}$ are positive integers and $Q_{n,i}\in M_{{n,i}}(C(Y_{n,i}))$ are projections. This theorem unifies and generalizes the reduction theorem for real rank zero $AH$ algebras due to Dadarlat and Gong ([D], [G3] and [DG]) and the reduction theorem for simple $AH$ algebras due to Gong (see [G4]).