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Abstract: Let A be an AH algebra, that is, A is the inductive limit -algebra of with , where are compact metric spaces, and are positive integers, and 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 . (This condition can be relaxed to a certain condition called very slow dimension growth.) In this article, we prove that if we further assume that is torsion free, then A is an approximate circle algebra (or an algebra), that is, A can be written as the inductive limit of where . One of the main technical results of this article, called the decomposition theorem, is proved for the general case, i.e., without the assumption that is torsion free. This decomposition theorem will play an essential role in the proof of a general reduction theorem, where the condition that is torsion free is dropped, in the subsequent paper Gong et al. (preprint) —of course, in that case, in addition to space , we will also need the spaces , , and , as in Gong (2002) . [Copyright &y& Elsevier]