L-stability in Rings and Left Quasi-duo Rings
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2018-05-28
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Abstract
A ring R is said to have stable range 1 if, for any element a in R and any left ideal L of R, Ra+L=R implies a-u in L for some unit u in R. Here we insist only that this holds for all L in some non-empty set L(R) of left ideals of R, and say that R is left L-stable in this case. We say that a class C of rings is affordable if C is the class of left L-stable rings for some left idealtor L. In addition to the rings of stable range 1, it is known that the left uniquely generated rings and the rings with internal cancellation are both affordable. Here, we explore L-stability in general, derive some properties of this phenomenon and show that it captures many well-known results. This in turn yields new information about the left uniquely generated rings and the internally cancellable rings, and enables us to answer some open questions related to them. More importantly, we show that the directly finite rings are affordable, which gives a new perspective on these rings and the plethora of open questions related to them. Next, we turn to the class of left quasi-duo rings, that is, rings R in which every maximal left ideal is an ideal. But, surprisingly, these rings have many nice natural characterizations and properties that have passed unnoticed since they were introduced in 1995. Here we discuss this class of rings and prove some interesting new results for them. In particular, we introduce the notion of left width for rings, and use it to give a characterization of any left quasi-duo ring that has a finite left width. After that, a characterization of the I-finite left quasi-duo rings will be given. Finally, we introduce the left-max idealtor which is defined in terms of the maximal left ideals for any ring R. Then we study and characterize its associated rings which will be called the left-max stable rings. We also show that the class of left quasi-duo rings is neither left-max stable nor affordable.
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Horoub, A. M. A, (2018). L-stability in Rings and Left Quasi-duo Rings (Doctoral thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca. doi:10.11575/PRISM/31958