mathcal{L}$-STABLE RINGS

If $\mathcal{L}(R)$ is a set of left ideals defined in any ring $R,$ we say that $R$ is $\mathcal{L}$-stable if it has stable range 1 relative to the set $\mathcal{L}(R)$. We explore $\mathcal{L}$-stability in general, characterize when it passes to related classes of rings, and explore which classes of rings are $\mathcal{L}$-stable for some$\mathcal{\ L}.$ Some well known examples of $\mathcal{L}$-stable rings are presented, and we show that the Dedekind finite rings are $\mathcal{L}$-stable for a suitable $\mathcal{L}$.