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Let $A$ be an $AH$ algebra, that is, $A$ is the inductive limit$C^{*}$-algebra of$$A_{1}\xrightarrow{\phi_{1,2}}A_{2}\xrightarrow{\phi_{2,3}}A_{3}\longrightarrow\cdots\longrightarrowA_{n}\longrightarrow\cdots$$ with$A_{n}=\bigoplus_{i=1}^{t_{n}}P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}$, 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 $A$ hasthe ideal property: each closed two-sided ideal of $A$ is generated by theprojections inside the ideal, as a closed two-sided ideal. Suppose that$\sup_{n,i}dim(X_{n,i})<+\infty$. In this article, we prove that $A$ can bewritten as the inductive limit of $$B_{1}\longrightarrowB_{2}\longrightarrow\cdots\longrightarrow B_{n}\longrightarrow\cdots,$$ where$B_{n}=\bigoplus_{i=1}^{s_{n}}Q_{n,i}M_{\{n,i\}}(C(Y_{n,i}))Q_{n,i}$, 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 $Q_{n,i}\inM_{\{n,i\}}(C(Y_{n,i}))$ are projections. This theorem unifies and generalizesthe 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 toGong (see [G4]).

A Reduction theorem for $AH$ algebras with ideal property