Summary: A \(C^*\)-algebra has the ideal property if any ideal (closed, two-sided) is generated (as an ideal) by its projections. We prove a theorem which implies, in particular, that an AH-algebra (AH stands for “approximately homogeneous”) stably isomorphic to a \(C^*\)-algebra with the ideal property has the ideal property. It is shown that, for any AH algebra \(A\) with the ideal property and slow dimension growth, the projections in \(M_\infty(A)\) satisfy the Riesz decomposition and interpolation properties and \(K_0(A)\) is a Riesz group. We prove a theorem which describes the partially ordered set of all the ideals generated by projections of an AH algebra \(A\); the special case when the projections in \(M_\infty(A)\) satisfy the Riesz decomposition property is also considered. This theorem generalizes a result of G.A. Elliot which gives the ideal structure of an AF algebra. We answer – jointly with M. Dadarlat – a question of G.K. Pedersen, constructing extensions of \(C^*\)-algebras with the ideal property which do not have the ideal property.