L-stable Rings and their Properties

Abstract

A unital ring R is called SR1 if for any element a 2 R and any left ideal L of R, Ra + L = R implies a 􀀀 u 2 L for some unit u in R. From this perspective, for some speci c set L(R) of left ideals of R, the condition still hold. These rings will be called then L-stable rings. For elements, an element a 2 R is L-stable if Ra+L = R, L 2 L(R), implies that a 􀀀 u 2 L for some unit u of R. Then R is an L-stable ring if each element of R is L-stable. A class C of rings is a orded by L if C = fL-stableg-the class of all L-stable rings, and C is a ordable if this happens for some certain set of left ideals of R. The class of all SR1 rings is a prototypical example of an a ordable class of rings. Some other well-known examples of L-stable rings are mentioned. It turns out that a ordable classes of rings share many interesting properties. i P lm 􀀀 A 􀀀Ð (SR1) 1 Ab «@ 􀀀Ð Tql Yms R ­d w 􀀀 􀀀Ð Tql 􀀀 Ra+L = R A 􀀀Ð ¢ Aqq § R L ©CAs§ ¨ A ¤ a ∈ R rOn Ay A m 􀀀 A wm {` , lWnm 􀀀 􀀀@¡ .u ∈ U(R) {`b a − u ∈ L £@¡ Yl lW wF .ªrK 􀀀 􀀀@¡ q t§ 􀀀z§ ¯ R Tql L(R) T§CAsy 􀀀 wq ,r}An`l Tbsn A A .(L−stable rings) L þ Tt A 􀀀 Aql A Aql 􀀀 ,Ra + L = R A 􀀀Ð (L−stable) L þ A a ∈ R Tql 􀀀 rOn` 􀀀 wk Tql 􀀀 ¨ At A ¤ .u ∈ U(R) {`b a − u ∈ L Y © ¥§ L ∈ L(R) C O 􀀀 wq .L þ A A¡r}An rOn A 􀀀Ð L þ Tt A C ¤ ,L þ Tt A 􀀀 Aql 􀀀 } −C = {L − stable} A 􀀀Ð L þ r t } .R T§CAsy 􀀀 Ay A m 􀀀 ­ d T wm d ¤ ry tl A .ry tl A Aql } Yl ¨FAF A w¡ 1 Ab 􀀀 «d 􀀀Ð Aql 􀀀 yb .A¡r Ð ry tl Tl A Aql wf} Yl T ¤r`m 􀀀 Tl ±􀀀 {` ­ry m 􀀀 AfO 􀀀 d§d` 􀀀 ¨ rtK ry tl Tl Aq 􀀀 Aql 􀀀 wf} ¢ . Amt¡®