primitive ideal space is $\widetilde{{\ensuremath{\mathbb{N}}\xspace}}$ and each fiber is either an AF-algebra or a purely infinite simple $C^{*}$-algebra. Here $\widetilde{{\ensuremath{\mathbb{N}}\xspace}} = {\ensuremath{\mathbb{N}}\xspace}\cup \{ \infty \}$ is the one-point compactification of ${\ensuremath{\mathbb{N}}\xspace}$. We show that an ordered isomorphism between ideal-related $K$-theory with coefficients (as defined in [@mdrm:ethy]) lifts to an isomorphism between the stabilized $C^{*}$-algebras. Moreover, if the $\infty$ fiber has torsion-free $K$-theory, then ideal-related $K$-theory with coefficients can be replace by ideal-related $K$-theory (which in general is a much simpler invariant). This is done by proving a universal coefficient theorem involving ideal-related $K$-theory for $C^{*}$-algebras over $\widetilde{{\ensuremath{\mathbb{N}}\xspace}}$ whose $\infty$ fiber has torsion-free $K$-theory. It was shown in [@mdrm:ethy] that a universal coefficient theorem involving ideal-related $K$-theory does not exist in general.

We note that we can not use the results in [@ERRshift] since the extension $$\begin{aligned}
0 \to \mathfrak{A}( {\ensuremath{\mathbb{N}}\xspace}) \to \mathfrak{A} \to \mathfrak{A} (\infty) \to 0\end{aligned}$$ does not satisfy the property that for every nonzero $a \in \mathfrak{A}( \infty )$, the ideal generated by $\tau ( a )$ in the corona algebra ${\mathcal{Q}( \mathfrak{A}({\ensuremath{\mathbb{N}}\xspace}) )\xspace}$ is ${\mathcal{Q}( \mathfrak{A}({\ensuremath{\mathbb{N}}\xspace}))\xspace}$. Here, $\tau$ denotes the Busby map of the above extension. Instead, we prove existence and uniqueness theorems, which together with an 