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¡®
