projections $p$ and $q$ in $\mathfrak{A}$ that generate the same closed ideal $\mathfrak{I}$ in $\mathfrak{A}$ and have the same image in $K_{0} ( \mathfrak{I} )$ must be Murray-von Neumann equivalent. If $\mathsf{M}_{n} ( \mathfrak{A} )$ has weak cancellation for every $n$, then we say that $\mathfrak{A}$ has *stable weak cancellation*.

Note that $\mathfrak{A}$ has stable weak cancellation if and only if $\mathfrak{A} \otimes {\ensuremath{\mathbb{K}}\xspace}$ has weak cancellation. Ara, Moreno, and Pardo in [@amp:nonstablekthy] showed that every graph $C^{*}$-algebra has stable weak cancellation. It is an open question if every real rank zero $C^{*}$-algebra has stable weak cancellation.

\[p:rrstwc\] Let $\mathfrak{A}$ be a $C^{*}$-algebra over $\widetilde{{\ensuremath{\mathbb{N}}\xspace}}$. If $\mathfrak{A}$ has real rank zero and $\mathfrak{A}(n)$ has stable weak cancellation for all $n \in \widetilde{{\ensuremath{\mathbb{N}}\xspace}}$, then $\mathfrak{A}$ has stable weak cancellation.

Since $\mathfrak{A}(n)$ has stable weak cancellation for each $n \in {\ensuremath{\mathbb{N}}\xspace}$, then $c_{0} ( \{ \mathfrak{A}(n) \} )$ has stable weak cancellation. The proposition now follows from [@err:fullext Lemma 3.15].

\[l:liftingfdquot\] Let $\mathfrak{A}$ be a $C^{*}$-algebra and let $\mathfrak{I}$ be an ideal of $\mathfrak{A}$ such that $\mathfrak{A} / \mathfrak{I}$ is a finite dimensional $C^{*}$-algebra. If for every projection $p\in \mathfrak A$ the corner $p\mathfrak{I}p$ has an approximate identity consisting of projections, and 