Abstract
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 $\mathcal{L}$.
Highlights
A ring R is said to have stable range 1 if, for any a ∈ R and left ideal L ⊆ R, Ra + L = R implies a − u ∈ L for some unit u in R
We show that the Dedekind finite rings arise as the set of L-stable rings for a suitable L, which gives a new perspective on these rings
We are interested in rings R of stable range 1 (SR1), that is every element a ∈ R is SR1 in the sense that a satisfies the following condition
Summary
Bass’ rings of stable range 1 arise if L(R) is the set of all left ideals in R, and it is known that Kaplansky’s uniquely generated rings and Ehrlich’s rings with internal cancellation arise in this way for other choices of L. We explore L-stability in general, derive some properties of this phenomenon, show that it captures many well known results, and characterize when L-stability passes to related rings. This in turn yields new information about left uniquely generated and internally cancellable rings. If each element of a set X ⊆ R has a ring property p, we say X has p
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