$\mathcal{L}$-STABLE RINGS

Year 2021, Volume: 29 Issue: 29, 63 - 94, 05.01.2021

Ayman M. A. Horoub W. K. Nıcholson

https://doi.org/10.24330/ieja.852012

Cited By: 1

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 $\mathcal{L}$.

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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