$mathcal{L}$-STABLE RINGS

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International Electronic Journal of Algebra
Volume:29 Issue:29
$mathcal{L}$-STABLE RINGS

$mathcal{L}$-STABLE RINGS

Authors : Ayman M A HOROUB, W K NICHOLSON

Pages : 63-94

Doi:10.24330/ieja.852012

View : 93 | Download : 9

Publication Date : 2021-01-05

Article Type : Research Paper

Abstract :If $\mathcal{L}insert ignore into journalissuearticles values(R);$ 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 $\mathcal{L}insert ignore into journalissuearticles values(R);$. 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 $\mathcal{L}$.
Keywords : Stable range, uniquely generated ring, internal cancellation ring, von Neumann regular ring, unit regular ring, triangular matrix ring, left idealtors, mathcal L stable ring

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