an $AH$ algebra with real rank zero, then [@EG] proves a decomposition result which says that $\phi_{m,n}$ can be approximately decomposed as a sum of two parts, $\phi_1$ and $\phi_2$; one part having a very small support projection, and the other part factoring through a finite dimensional algebra.

In this article, the basic building blocks we consider were introduced by Elliott in [@Ell3] and Thomsen in [@ET], which are also called one dimensional non-commutative finite CW complexes. The ${\mathrm{C}}^*$-algebras we consider are the algebras which can be expressed as the real rank zero inductive limit of a sequence $$A_1\,\xrightarrow{\phi_{1,2}}\,A_2\,\xrightarrow{\phi_{2,3}}\,A_3\rightarrow\cdots$$ with $A_n=\oplus_{i=1}^{n_i}A_{[n,i]}$, where all the $A_{[n,i]}$ are Elliott-Thomsen building blocks and $\phi_{n,n+1}$ are homomorphisms. It will be shown that many inductive limit algebras are within the above class. We will prove a decomposition result for these algebras, which will be used in the proof of uniqueness theorem and of the existence theorem.

This paper is organized as follows. In Section 1, we will introduce some notations, and collect some known results. In Section 2, we use the generalized pairing lemma to get some comparable results about the spectra. In Section 3, we will prove the main decomposition result.

Preliminaries
=============

Let 