Quasi-unit regularity and QB-rings
Some relations for quasiunit regular rings and QB-rings, as well as for pseudounit regular rings and QB ∞-rings, are obtained. In the first part of the paper, we prove that (an exchange ring R is a QB-ring) ⟺ (whenever x ∈ R is regular, there exists a quasiunit regular element w ∈ R such that x = xy...
Saved in:
| Published in: | Український математичний журнал |
|---|---|
| Date: | 2012 |
| Main Authors: | , , , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2012
|
| Subjects: | |
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/164158 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Quasi-unit regularity and QB-rings/ Jianghua Li, Xiaoqing Sun, Xiaoqin Shen, Shangping Wang // Український математичний журнал. — 2012. — Т. 64, № 3. — С. 415-425. — Бібліогр.: 9 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-164158 |
|---|---|
| record_format |
dspace |
| spelling |
Jianghua Li Xiaoqing Sun Xiaoqin Shen Shangping Wang 2020-02-08T17:14:26Z 2020-02-08T17:14:26Z 2012 Quasi-unit regularity and QB-rings/ Jianghua Li, Xiaoqing Sun, Xiaoqin Shen, Shangping Wang // Український математичний журнал. — 2012. — Т. 64, № 3. — С. 415-425. — Бібліогр.: 9 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/164158 512.5 Some relations for quasiunit regular rings and QB-rings, as well as for pseudounit regular rings and QB ∞-rings, are obtained. In the first part of the paper, we prove that (an exchange ring R is a QB-ring) ⟺ (whenever x ∈ R is regular, there exists a quasiunit regular element w ∈ R such that x = xyx = xyw for some y ∈ R) ⟺ (whenever aR + bR = dR in R; there exists a quasiunit regular element w ∈ R such that a + bz = dw for some z ∈ R). Similarly, we also give necessary and sufficient conditions for QB ∞-rings in the second part of the paper. Отримано деякi спiввiдношення для квазiодиничних регулярних кiлець та QB-кiлець, а також для псевдоодиничних регулярних кiлець та QB∞-кiлець. У першiй частинi статтi доведено, що (кiльце R з властивiстю замiни є QB-кiльцем) ⇔ (якщо x∈R є регулярним, то iснує квазiодиничний регулярний елемент w∈R такий, що x=xyx=xyw для деякого y∈R) ⇔ (якщо aR+bR=dR in R в R, то iснує квазiодиничний регулярний елемент w∈R такий, що a+bz=dw для деякого z∈R). Аналогiчним чином отриманi необхiднi та достатнi умови для QB∞-кiлець наведено у другiй частинi статтi. This paper is supported by National Nature Science Foundation of China (NSFC 61173192, 11101330) and Natural Science Foundation of Shaanxi Province (2011JQ1007) and Education Office Foundation of Shaanxi Province (2010JK728) and The Starting Research Fund from Xi’an University of Technology (108-211105). en Інститут математики НАН України Український математичний журнал Статті Quasi-unit regularity and QB-rings Квазiодинична регулярнiсть та QB-кiльця Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Quasi-unit regularity and QB-rings |
| spellingShingle |
Quasi-unit regularity and QB-rings Jianghua Li Xiaoqing Sun Xiaoqin Shen Shangping Wang Статті |
| title_short |
Quasi-unit regularity and QB-rings |
| title_full |
Quasi-unit regularity and QB-rings |
| title_fullStr |
Quasi-unit regularity and QB-rings |
| title_full_unstemmed |
Quasi-unit regularity and QB-rings |
| title_sort |
quasi-unit regularity and qb-rings |
| author |
Jianghua Li Xiaoqing Sun Xiaoqin Shen Shangping Wang |
| author_facet |
Jianghua Li Xiaoqing Sun Xiaoqin Shen Shangping Wang |
| topic |
Статті |
| topic_facet |
Статті |
| publishDate |
2012 |
| language |
English |
| container_title |
Український математичний журнал |
| publisher |
Інститут математики НАН України |
| format |
Article |
| title_alt |
Квазiодинична регулярнiсть та QB-кiльця |
| description |
Some relations for quasiunit regular rings and QB-rings, as well as for pseudounit regular rings and QB ∞-rings, are obtained. In the first part of the paper, we prove that (an exchange ring R is a QB-ring) ⟺ (whenever x ∈ R is regular, there exists a quasiunit regular element w ∈ R such that x = xyx = xyw for some y ∈ R) ⟺ (whenever aR + bR = dR in R; there exists a quasiunit regular element w ∈ R such that a + bz = dw for some z ∈ R). Similarly, we also give necessary and sufficient conditions for QB ∞-rings in the second part of the paper.
Отримано деякi спiввiдношення для квазiодиничних регулярних кiлець та QB-кiлець, а також для псевдоодиничних регулярних кiлець та QB∞-кiлець. У першiй частинi статтi доведено, що (кiльце R з властивiстю замiни є QB-кiльцем) ⇔ (якщо x∈R є регулярним, то iснує квазiодиничний регулярний елемент w∈R такий, що x=xyx=xyw для деякого y∈R) ⇔ (якщо aR+bR=dR in R в R, то iснує квазiодиничний регулярний елемент w∈R такий, що a+bz=dw для деякого z∈R). Аналогiчним чином отриманi необхiднi та достатнi умови для QB∞-кiлець наведено у другiй частинi статтi.
|
| issn |
1027-3190 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/164158 |
| citation_txt |
Quasi-unit regularity and QB-rings/ Jianghua Li, Xiaoqing Sun, Xiaoqin Shen, Shangping Wang // Український математичний журнал. — 2012. — Т. 64, № 3. — С. 415-425. — Бібліогр.: 9 назв. — англ. |
| work_keys_str_mv |
AT jianghuali quasiunitregularityandqbrings AT xiaoqingsun quasiunitregularityandqbrings AT xiaoqinshen quasiunitregularityandqbrings AT shangpingwang quasiunitregularityandqbrings AT jianghuali kvaziodiničnaregulârnistʹtaqbkilʹcâ AT xiaoqingsun kvaziodiničnaregulârnistʹtaqbkilʹcâ AT xiaoqinshen kvaziodiničnaregulârnistʹtaqbkilʹcâ AT shangpingwang kvaziodiničnaregulârnistʹtaqbkilʹcâ |
| first_indexed |
2025-11-25T22:57:34Z |
| last_indexed |
2025-11-25T22:57:34Z |
| _version_ |
1850576478310236160 |
| fulltext |
UDC 512.5
Xiaoqing Sun, Shangping Wang, Xiaoqin Shen, Jianghua Li
(School Sci., Xi’an Univ. Technology, China)
QUASI-UNIT REGULARITY AND QB-RINGS*
КВАЗIОДИНИЧНА РЕГУЛЯРНIСТЬ ТА QB-КIЛЬЦЯ
Some relations for quasiunit regular rings and QB-rings, as well as for pseudounit regular rings and QB∞-rings, are
obtained. In the first part of the paper, we prove that (an exchange ring R is a QB-ring) ⇔ (whenever x ∈ R is regular,
there exists a quasiunit regular element w ∈ R such that x = xyx = xyw for some y ∈ R)⇔ (whenever aR+ bR = dR
in R, there exists a quasiunit regular element w ∈ R such that a + bz = dw for some z ∈ R). Similarly, we also give
necessary and sufficient conditions for QB∞-rings in the second part of the paper.
Отримано деякi спiввiдношення для квазiодиничних регулярних кiлець та QB-кiлець, а також для псевдоодиничних
регулярних кiлець та QB∞-кiлець. У першiй частинi статтi доведено, що (кiльце R з властивiстю замiни є QB-
кiльцем) ⇔ (якщо x ∈ R є регулярним, то iснує квазiодиничний регулярний елемент w ∈ R такий, що x = xyx =
= xyw для деякого y ∈ R)⇔ (якщо aR+ bR = dR в R, то iснує квазiодиничний регулярний елемент w ∈ R такий,
що a + bz = dw для деякого z ∈ R). Аналогiчним чином отриманi необхiднi та достатнi умови для QB∞-кiлець
наведено у другiй частинi статтi.
1. Introduction. Let R be an associative ring with nonzero identity. Recall that a ring R is an
exchange ring if for every right R-module A and any decomposition A = M ′ ⊕ N =
⊕
i∈I Ai,
where M ′R ' RR and the index set I is finite, there exist submodules A′i ⊆ Ai such that A =
= M ′
⊕
(
⊕
i∈I A
′
i) [8]. The class of exchange rings is large and includes all von Neumann regular
rings, all π-regular rings and C∗-algebras of real rank zero [1] etc. The ring R is said to have stable
range one provided that whenever ax+ b = 1 in R, there exists y ∈ R such that a+ by is a unit in
R. An exchange ring R has stable range one if and only if whenever x ∈ R is regular, there exists
a unit-regular element w ∈ R such that x = xyx = xyw for some y ∈ R if and only if whenever
aR + bR = dR in R, there exists a unit regular element w ∈ R such that a + bz = dw for some
z ∈ R [9]. Some necessary and sufficient conditions under which an exchange ring R has weakly
stable range one are also proved.
Replacing invertibility with quasi-invertibility in stable range one Pere Ara discover a new class
of rings, the QB-rings [2]. The ring R is a QB-ring provided whenever aR + bR = R in R, there
exists y ∈ R such that a + by is quasi-invertible in R. As well known, this definition is left-right
symmetric. Replacing R−1q with R−1∞ in the definition of QB-ring, we say that a ring is QB∞-ring
if whenever aR+ bR = R in R, there exists y ∈ R such that a+ by ∈ R−1∞ [6].
In this paper, the definitions of quasi-unit regular and pseudo-unit regular are given. An element
x ∈ R is called quasi-unit regular (pseudo-unit regular) if there exists a quasi-invertible (pseudo-
invertible) element u ∈ R such that x = xux. The purpose of this article is to investigate the
relations of quasi-unit regular and QB-rings, as well as pseudo-unit regular and QB∞-rings. It is
shown in Section 2 that an exchange ring R is a QB-ring if and only if whenever x ∈ R is regular,
there exists a quasi-unit regular element w ∈ R such that x = xyx = xyw for some y ∈ R if and
*This paper is supported by National Nature Science Foundation of China (NSFC 61173192, 11101330) and Natural
Science Foundation of Shaanxi Province (2011JQ1007) and Education Office Foundation of Shaanxi Province (2010JK728)
and The Starting Research Fund from Xi’an University of Technology (108-211105).
c© XIAOQING SUN, SHANGPING WANG, XIAOQIN SHEN, JIANGHUA LI, 2012
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3 415
416 XIAOQING SUN, SHANGPING WANG, XIAOQIN SHEN, JIANGHUA LI
only if for any regular x ∈ R there exist a quasi-unit regular element w ∈ R and an idempotent
e ∈ R such that x = ew if and only if whenever aR + bR = dR in R, there exists a quasi-unit
regular element w such that a + bz = dw for some z ∈ R. In Section 3, we extend these to QB∞-
ring. It is extended the results of Chen [7]. We prove that an exchange ring R is a QB∞-ring if
and only if whenever x ∈ R is regular, there exists a pseudo-unit regular element w ∈ R such that
x = xyx = xyw for some y ∈ R.
Throughout this paper, R denotes an associative ring with identity. We denote by R−1, E(R)
the set of all units of R, the set of all idempotents in R, respectively. An element x ∈ R is regular
provided that x = xyx for some y ∈ R, which is also commonly known as von Neumann regular.
2. Quasi-unit regular. Let us start by recalling the concept of quasi-invertibility. We say that
elements x and y in a ring R are centrally orthogonal provided that xRy = yRx = 0, and we write
x⊥y. An element u in an arbitrary ring R is said to be quasi-invertible if there exist elements a, b
in R such that
(1− ua)⊥(1− bu). (2.1)
The set of quasi-invertible elements in R will be denoted by R−1q . It is easily checked that R−1R−1q =
= R−1q and R−1q R−1 = R−1q .
If u ∈ R−1q , then we have the equation (1 − ua)u(1 − bu) = 0. Taking v = a + b − aub this
implies that u = uvu. By computation 1−uv = (1−ua)(1− bu) and 1−vu = (1−au)(1−ub), so
that we have the relation (1− uv)⊥(1− vu). We say in this situation that v is a quasi-inverse of u.
Definition 2.1. Let R be a ring. An element x ∈ R is quasi-unit regular if there exists a
quasi-invertible element u ∈ R such that x = xux. A ring R is quasi-unit regular if every element
in R is quasi-unit regular.
Lemma 2.1. Let R be a ring and x ∈ R. Then the following are equivalent:
(1) x is quasi-unit regular;
(2) x = xyx = xyu, where y, u ∈ R and u ∈ R−1q ;
(2′) x = xyx = uyx, where y, u ∈ R and u ∈ R−1q ;
(3) x = xyx = xyw, where y, w ∈ R and w is quasi-unit regular;
(3′) x = xyx = wyx, where y, w ∈ R and w is quasi-unit regular.
Proof. (1) ⇒ (2). Since x is quasi-unit regular, there exists a quasi-invertible element u ∈ R
such that x = xux. Let ux = e and 1− xu = f . Then e, f ∈ E(R) and
euxu+ uf = uxuxu+ u(1− xu) = u, e(uxu+ uf) + (1− e)uf = u.
Let g = (1− e)ufu−1q (1− e) where u−1q is the quasi-inverse of u. Since (1− e)uf = (1− e)u, we
have
g2 = g, (1− e)u = (1− e)uu−1q (1− e)u = g(1− e)u = gu.
Therefore
u(x+ fu−1q (1− e))(1− eufu−1q (1− e))u = (ux+ ufu−1q (1− e))(1− eufu−1q (1− e))u =
= (e+ ufu−1q (1− e))(1− eufu−1q (1− e))u = (e(1− eufu−1q (1− e)) + ufu−1q (1− e))u =
= (e+ (1− e)ufu−1q (1− e))u = (e+ g)u = u.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
QUASI-UNIT REGULARITY AND QB-RINGS 417
Let
v = (1− eufu−1q (1− e))u = (1 + eufu−1q (1− e))−1u, p = x+ fu−1q (1− e).
Then vpv = v. Since R−1R−1q = R−1q = R−1q R−1, we have v ∈ R−1q .
Since (1− v−1q v)R(1− vv−1q ) = 0, we have (1− v−1q v)p(1− vv−1q ) = 0. Then p = v−1q + 2p−
− v−1q vp − pvv−1q = v−1q + (1 − v−1q v)p + p(1 − vv−1q ). In view of Theorem 2.3 [2], we conclude
that p ∈ R−1q . It is clear that
x = xux = xu(x+ fu−1q (1− e)) = xup.
(2) ⇒ (1). Suppose that x = xyx = xyu where u ∈ R−1q . Let z = yxy. Then x = xzx = xzu
and z = zxz. Hence z = z(x + (1 − xz)u)z where x + (1 − xz)u = u ∈ R−1q . That is, z is
quasi-unit regular. It follows from (1) ⇒ (2) that there exists a p ∈ R−1q such that z = zuz = zup.
Let e = 1− zx and f = zu. Then e, f ∈ E(R) and
fpx(1− f) + e(1− f) = 1− f, (1− f)e(1− f) = 1− f.
Then
z + e(1− f)p = fp+ e(1− f)p = (1 + fpx(1− f))−1p ∈ R−1q .
It is clear that x = x(z + e(1− f)p)x with z + e(1− f)p ∈ R−1q . Therefore, x is quasi-unit regular.
(2)⇒ (3). It is trivial.
(3) ⇒ (2). Let x = xyx = xyw where w is quasi-unit regular. It follows from (1) ⇒ (2), we
have w = ep where e2 = e and p ∈ R−1q . It follows from the equation xy + (1 − xy) = 1 we have
xyw+ (1−xy)w = w. Since x = xyw, we have x+ (1−xy)w = w. Then xy+ (1−xy)wy = wy.
Hence wy + (1− xy)(1−wy) = 1. It follows that ewy(1− e) + (1− xy)(1−wy)(1− e) = 1− e.
Consequently,
e+ (1− xy)(1− wy)(1− e) = 1− ewy(1− e) = (1 + ewy(1− e))−1
is invertible in R. Let
u = w + (1− xy)(1− wy)(1− e)p = (e+ (1− xy)(1− wy)(1− e))p.
Since R−1R−1q = R−1q and R−1q R−1 = R−1q , we have u ∈ R−1q . It is easy to check that x = xyx =
= xyw = xyu where u ∈ R−1q .
Similarly, we can prove equivalences of (1), (2′), (3′).
Lemma 2.1 is proved.
Corollary 2.1. Let R be a ring and x ∈ R be regular. Then the following are equivalent:
(1) x is quasi-unit regular;
(2) there exist some idempotent e ∈ R and some quasi-invertible element u ∈ R such that
x = eu;
(2′) there exist some idempotent e ∈ R and some quasi-invertible element u ∈ R such that
x = ue;
(3) there exist some idempotent e ∈ R and some quasi-unit regular element w ∈ R such that
x = ew;
(3′) there exist some idempotent e ∈ R and some quasi-unit regular element w ∈ R such that
x = we.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
418 XIAOQING SUN, SHANGPING WANG, XIAOQIN SHEN, JIANGHUA LI
Proof. (1)⇒ (2). It follows from (1)⇒ (2) of Lemma 2.1.
(2)⇒ (3). It is obvious.
(3)⇒ (1). Assume x = xyx = ew, where e ∈ R is an idempotent and w is quasi-unit regular. Let
w = wuw where u is a quasi-invertible in R. Since xy+(1−xy) = 1, we have ewy+(1−xy) = 1.
It follows that
ewy(1− e) + (1− xy)(1− e) = 1− e.
Then
v := e+ (1− xy)(1− e) = 1− ewy(1− e) = (1 + ewy(1− e))−1
is a unit in R. Let
p = x+ (1− xy)(1− e)w = (e+ (1− xy)(1− e))w = vw = vwuw = vw(uv−1)vw.
Since R−1R−1q = R−1q and R−1q R−1 = R−1q , we have uv−1 ∈ R−1q . Then q is quasi-unit regular. It is
easy to check that x = xyx = xy(x+(1−xy)(1−e)w) = xyp. The result follows from Lemma 2.1.
Similarly, we can prove equivalences of (1), (2′), (3′).
Corollary 2.1 is proved.
By the result of Theorem 8.4 [2], an exchange ring R is a QB-ring if and only if every regular
element in R is quasi-unit regular. It follows from Lemma 2.1, we immediately have the following
characterizations of exchange QB-ring.
Theorem 2.1. Let R be an exchange ring. Then the following are equivalent:
(1) R is a QB-ring;
(2) whenever x ∈ R is regular, there exists a u ∈ R−1q such that x = xyx = xyu for some
y ∈ R;
(2′) whenever x ∈ R is regular, there exists a u ∈ R−1q such that x = xyx = uyx for some
y ∈ R;
(3) whenever x ∈ R is regular, there exists a quasi-unit regular element w ∈ R such that
x = xyx = xyw for some y ∈ R;
(3′) whenever x ∈ R is regular, there exists a quasi-unit regular element w ∈ R such that
x = xyx = wyx for some y ∈ R.
By Theorem 2.1, an exchange ring R is a QB-ring if and only if whenever x = xyx ∈ R, there
exists a quasi-invertible element u ∈ R such that x = xyu if and only if whenever x = xyx ∈ R,
there exists a quasi-invertible element u ∈ R such that x = uyx. The following theorem gives a
common quasi-invertible element u ∈ R such that x = xyu = uyx.
Theorem 2.2. Let R be an exchange ring. Then the following are equivalent:
(1) R is a QB-ring;
(2) whenever x = xyx, there exists a quasi-invertible element u ∈ R such that x = xyu = uyx;
(3) whenever x = xyx, there exists a quasi-invertible element u ∈ R such that xyu = uyx.
Proof. (1) ⇒ (2). For any x = xyx in R, we have x = xzx and z = zxz with z = yxy. By
Theorem 8.4 [2], we have z = zxz = zvz for some quasi-invertible element v ∈ R. Let
u = (1− xz − vz)v(1− zx− zv) = v − vzv + x.
Since v ∈ R−1q , there exist a, b ∈ R such that (1− va)⊥(1− bv). It is easily checked that (1− xz −
− vz)2 = 1 and (1− zx− zv)2 = 1. Then
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
QUASI-UNIT REGULARITY AND QB-RINGS 419
(1− u(1− zx− zv)a(1− xz − vz)) = (1− xz − vz)(1− va)(1− xz − vz),
(1− (1− zx− zv)b(1− xz − vz)u) = (1− zx− zv)(1− bv)(1− zx− zv).
Hence, (1− u(1− zx− zv)a(1− xz − vz))⊥(1− (1− zx− zv)b(1− xz − vz)u). Therefore, u is
quasi-invertible. It follows from
xzu = xzv − xzvzv + xzx = xzx = x, uzx = vzx− vzvzx+ xzx = xzx = x
we obtain that x = xyu = xzu = uzx = uyx with u ∈ R−1q .
(2)⇒ (3). It is obvious.
(3) ⇒ (1). For any x = xyx in R, there exists a quasi-invertible element u ∈ R such that
xyu = uyx. Define
η : xyR = xR ' yxR, r ∈ R, η(xr) = yxr;
α : (1− xy)R→ (1− yx)R, r ∈ R, (1− xy)r → (1− yx)u−1q (1− xy)r;
β : (1− yx)R→ (1− xy)R, r ∈ R, (1− yx)r → (1− xy)ur.
Since (1 − xy)u = u(1 − yx), we easily check that α and β are right R-module homomorphisms.
Define
φ : R = xR⊕ (1− xy)R→ yxR⊕ (1− yx)R = R,
x1 ∈ xR, x2 ∈ (1− xy)R, φ(x1 + x2) = η(x1) + α(x2);
ψ : R = yxR⊕ (1− yx)R→ xR⊕ (1− xy)R = R,
y1 ∈ yxR, y2 ∈ (1− yx)R, ψ(y1 + y2) = η−1(y1) + β(y2).
Then
(1− ψφ)(x1 + x2) = x2 − (1− xy)u−1q ux2 =
= (1− xy)x2 − (1− xy)u−1q ux2 = (1− xy)(1− u−1q u)x2
for any x1 ∈ xR, x2 ∈ (1− xy)R. On the other hand,
(1− φψ)(y1 + y2) = y2 − (1− yx)uu−1q y2 = (1− yx)(1− uu−1q )y2
for any y1 ∈ yxR, y2 ∈ (1−yx)R. Then we have φ is quasi-invertible such that x = xφx. Therefore
R is a QB-ring.
Theorem 2.2 is proved.
Chen had shown that an exchange ring R is a QB-ring if and only if for any regular x ∈ R,
there exist e ∈ E(R) and u ∈ R−1q such that x = eu [5] (Theorem 5). Using Corollary 2.1, we have
following corollary.
Corollary 2.2. Let R be an exchange ring. Then the following are equivalent:
(1) R is a QB-ring;
(2) whenever x ∈ R is regular, there exists an idempotent e ∈ R and a quasi-unit regular element
w ∈ R such that x = ew;
(2′) whenever x ∈ R is regular, there exists an idempotent e ∈ R and a quasi-unit regular
element w ∈ R such that x = we.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
420 XIAOQING SUN, SHANGPING WANG, XIAOQIN SHEN, JIANGHUA LI
Canfell showed that R has stable range one if and only if aR+ bR = R implies that there exists
a unit u ∈ R such that a+ by = du for some y ∈ R, by using the method of completion of diagrams
[4] (Theorem 2.9). We generalize Canfell’s result to QB-rings.
Proposition 2.1. Let R be a ring. Then the following are equivalent:
(1) R is a QB-ring;
(2) whenever aR+ bR = R, there exists some z ∈ R such that a+ bz is quasi-invertible;
(3) whenever aR + bR = dR, there exists some quasi-invertible element u ∈ R such that
a+ bz = du for some z ∈ R.
Proof. (3)⇒ (2)⇒ (1) are obvious.
(1)⇒ (3). Let aR + bR = dR. Then a, b ∈ dR. Hence we may assume that a = dr and b = ds
for some r, s ∈ R. Let ax + by = d. Equivalently we have drx + dsy = d. It follows that dg = 0
where g = 1− rx− sy. Now from the fact that rx+ sy + g = 1 we have there exists some z′ ∈ R
such that r + (sy + g)z′ = u ∈ R−1q . Hence
du = d(r + (sy + g)z′) = a+ byz′ + dgz′ = a+ byz′ = a+ bz
where z = yz′.
Proposition 2.1 is proved.
In case R is an exchange ring. We even have the following more general result.
Theorem 2.3. Let R be an exchange ring. Then the following are equivalent:
(1) R is a QB-ring;
(2) whenever aR + bR = R, there exists some quasi-unit regular element w ∈ R such that
a+ bz = w for some z ∈ R;
(3) whenever aR + bR = dR, there exists some quasi-unit regular element w ∈ R such that
a+ bz = dw for some z ∈ R.
Proof. (1)⇒ (3). It follows from Proposition 2.1.
(3)⇒ (2). It is obvious.
(2) ⇒ (1). Let x = xyx for some y ∈ R. Since xy + (1 − xy) = 1. By assumptions we have
x+ (1− xy)z = w is quasi-unit regular for some z ∈ R. Hence
x = xyx = xy(w − (1− xy)z) = xyw.
The conclusion follows from Theorem 2.1.
Theorem 2.3 is proved.
Following a similar route above we give the following characterizations of QB-ring.
Theorem 2.4. Let R be an exchange ring. Then the following are equivalent:
(1) R is a QB-ring;
(2) whenever aR+bR = R, there exists a quasi-unit regular element w ∈ R such that aw+by = 1
for some y ∈ R;
(3) whenever aR + bR = R, there exist quasi-unit regular elements w1, w2 ∈ R such that
aw1 + bw2 = 1;
(4) whenever a1R + . . . + amR = R, there exist quasi-unit regular elements w1, . . . , wm ∈ R
such that aw1 + . . .+ amwm = 1, where m ≥ 2;
(5) whenever aR+bR = dR, there exists a quasi-unit regular element w ∈ R such that aw+by =
= d for some y ∈ R;
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
QUASI-UNIT REGULARITY AND QB-RINGS 421
(6) whenever aR + bR = dR, there exist quasi-unit regular elements w1, w2 ∈ R such that
aw1 + bw2 = d;
(7) whenever a1R + · · ·+ amR = dR, there exist quasi-unit regular elements w1, . . . , wm ∈ R
such that aw1 + . . .+ amwm = d, where m ≥ 2.
Proof. (7) ⇒ (4) ⇒ (3) ⇒ (2) and (7)⇒ (6) ⇒ (5)⇒ (2) are obvious.
(1) ⇒ (7). Assume that a1R + . . . + amR = dR. Then ai ∈ dR, i = 1, . . . ,m. Let ai = dti,
i = 1, . . . ,m. Obviously we have dt1x1+. . .+dtmxm = d for some xi ∈ R, i = 1, . . . ,m. It follows
that dg = 0, where g = 1 − (dt1x1 + . . . + dtmxm). Since t1x1 + . . . + tmxm + g = 1 we obtain
that t1R+ . . .+ tmR+ gR = R. Note that R is an exchange ring, so there exist idempotent ei ∈ R,
i = 1, . . . ,m, and idempotent f ∈ R, where ei and f are orthogonal satisfying e1 + . . .+em +f = 1
such that ei = tiyi, i = 1, . . . ,m, and f = gz for some yi, z ∈ R, i = 1, . . . ,m. Let wi = yiei,
i = 1, . . . ,m. Then tiwi = tiyiei = ei and witiwi = yieiei = yiei = wi. Since R is a QB-ring,
we have wi is quasi-unit regular by Theorem 8.4 [2]. It follows from t1w1 + . . . + tmwm + gz =
= e1 + . . .+ em + f = 1 that aw1 + . . .+ amwm = d(t1w1 + . . .+ tmwm + gz) = d.
(2)⇒ (1). Let x = xyx for some y ∈ R. Since yx+(1−yx) = 1, we have yR+(1−yx)R = R.
By assumptions there exists a quasi-unit regular element w ∈ R such that yw + (1 − yx)z = 1 for
some z ∈ R. Hence x = xyx = x(yw + (1− yx)z) = xyw. It follows from Theorem 2.1 that R is
a QB-ring.
Theorem 2.4 is proved.
The following proposition may be viewed as a supplement of Theorem 2.4 in case m = 1, which
also generalizes Theorem 4 [5].
Proposition 2.2. Let R be an exchange ring. Then the following are equivalent:
(1) R is a QB-ring;
(2) whenever aR = bR, there exists a quasi-invertible element u ∈ R such that b = au;
(3) whenever aR = bR, there exists a quasi-unit regular element w ∈ R such that b = aw.
Proof. (1)⇒ (2). Given aR = bR, then a = bx and b = ay for x, y ∈ R. From xy+(1−xy) =
= 1, we have z ∈ R such that x + (1 − xy)z = u ∈ R−1q . It is easy to verify that bxy = b. Then
a = bx = b(x+ (1− xy)z) = bu.
(2)⇒ (3). It is trivial.
(3) ⇒ (1). Let x = xyx for some y ∈ R. Since xR = xyR, we can find a quasi-unit regular
element w ∈ R such that x = xyw. Then x = xyx = xyw. It follows from Theorem 2.1 that R is a
QB-ring.
Proposition 2.2 is proved.
Corollary 2.3. Let R be an exchange ring. Then the following are equivalent:
(1) R is a QB-ring;
(2) whenever ψ : aR ' bR, where a, b ∈ R, there exists a quasi-invertible element u ∈ R such
that ψ(a) = bu;
(3) whenever ψ : aR ' bR, where a, b ∈ R, there exists a quasi-unit regular element w ∈ R such
that ψ(a) = bw.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
422 XIAOQING SUN, SHANGPING WANG, XIAOQIN SHEN, JIANGHUA LI
Proof. (1) ⇒ (2). If ψ : aR ' bR, then b = ψ(ax) and a = ψ−1(by) for some x, y ∈ R. Then
b = ψ(ax) = ψ(ψ−1(by)x) = byψ(x). Since yψ(x) + (1 − yψ(x)) = 1 and R is a QB-ring, we
have y + (1− yψ(x))z = u ∈ R−1q . Hence ψ(a) = by = b(y + (1− yψ(x))z) = bu.
(2) ⇒ (3). It is trivial.
(3) ⇒ (1). It follows from Proposition 2.2.
Corollary 2.3 is proved.
The ideas of the following result come from Lemma 1.2 [3].
Proposition 2.3. Let R be an exchange ring. Then the following are equivalent:
(1) R is a QB-ring;
(2) whenever x = xyx, there exists a ∈ R such that y − a is quasi-invertible and 1 − xa is
invertible;
(3) whenever x = xyx, there exists a ∈ R such that x − a is quasi-unit regular and 1 − ya is
invertible.
Proof. (1)⇒ (2). Let x = xyx for some y ∈ R. Since yx+(1−yx) = 1 and R is a QB-ring, we
have there exists some z ∈ R such that u : = y+ (1− yx)z is quasi-invertible. Let a = −(1− yx)z.
Then y − a = u. Moreover, since x = xyx, we have 1− xa = 1 + x(1− yx)z = 1 is invertible.
(2) ⇒ (3). Assume x = xyx. Let z = yxy. Obviously, x = xzx and z = zxz. By assumption,
there exists a′ ∈ R such that u := x− a′ is quasi-invertible and 1− za′ is invertible. Let a = xya′.
Then
1− ya = 1− yxya′ = 1− za′, x− a = xyx− xya′ = xy(x− a′) = eu,
where e = xy is an idempotent and u ∈ R−1q . Hence x− a is quasi-unit regular by Corollary 2.1.
(3) ⇒ (1). For any x = xyx in R, we have x = xzx and z = zxz with z = yxy. Then there
exists a′ ∈ R such that w := x− a′ is quasi-unit regular and u := 1− za′ is invertible. Hence
xyw = xy(x− a′) = x− xya′ = x− xyxya′ = x− xza′ = x(1− za′) = xu.
It follows that x = xywu−1 = xyw′ where w′ = wu−1. Assume that w = wpw, where p is the
quasi-invertible in R. Then
w′ = wu−1 = wpwu−1 = (wu−1)(up)(wu−1) = w′(up)w′,
where up ∈ R−1R−1q = R−1q . Therefore, we have x = xyx = xyw′ with w′ is quasi-unit regular. It
follows from Theorem 2.1 that R is a QB-ring.
Proposition 2.3 is proved.
3. Pseudo-unit regular. Recall that two elements x, y ∈ R are centrally orthogonal, denoted by
x⊥y, if xRy = 0 = yRx. We say that two elements x, y ∈ R are pseudo-orthogonal, denoted by
x\y, if RxRyR is nilpotent. Let R−1∞ =
{
u ∈ R | ∃ a, b ∈ R such that (1 − ua)\(1 − bu)
}
. It is
also easily checked that R−1R−1∞ = R−1∞ and R−1∞ R−1 = R−1∞ .
A ring R is a QB∞-ring provided that aR + bR = R implies that there exists y ∈ R such that
a+ by ∈ R−1∞ . Obviously, every QB-ring is a QB∞-ring.
Definition 3.1. Let R be a ring. An element x ∈ R is pseudo-unit regular if there exists
u ∈ R−1∞ such that x = xux. A ring R is pseudo-unit regular if every element in R is pseudo-unit
regular.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
QUASI-UNIT REGULARITY AND QB-RINGS 423
Lemma 3.1. Let R be a ring and x ∈ R. Then the following are equivalent:
(1) x is pseudo-unit regular;
(2) x = xyx = xyu, where u, y ∈ R and u ∈ R−1∞ ;
(2′) x = xyx = uyx, where u, y ∈ R and u ∈ R−1∞ ;
(3) x = xyx = xyw, where w, y ∈ R and w is pseudo-unit regular;
(3′) x = xyx = wyx, where w, y ∈ R and w is pseudo-unit regular.
Proof. (1) ⇒ (2). Since x is pseudo-unit regular, there exists u ∈ R−1∞ such that x = xux. Let
ux = e and 1 − xu = f . Then e2 = uxux = ux = e and f2 = (1 − xu)(1 − xu) = 1 − xu = f .
Hence euxu+ uf = uxuxu+ u(1− xu) = u and e(uxu+ uf) + (1− e)uf = u. Since u ∈ R−1∞ ,
there exists v ∈ R such that (1 − uv)\(1 − vu) and (R(u − uvu)R)m = 0 = (R(v − vuv)R)m for
some m ∈ N by Lemma 2.1 [6]. Let g = (1 − e)ufv(1 − e). Since (1 − e)uf = (1 − e)u, we see
that
(1− e)ufv(1− e))u = (1− e)uvu− (1− e)uveu = (1− e)uvu− (1− e)uvuxu =
= (1− e)uvu+ (1− e)(u− uvu)xu− (1− e)uxu =
= (1− e)uvu− (1− e)(u− uvu) + (1− e)(u− uvu)xu.
As a result, (1− e)u ≡ (1− e)ufv(1− e)u ≡ gu (mod R(u− uvu)R). Similarly, we have
g2 ≡ (1− e)ufv(1− e)ufv(1− e) ≡ (1− e)ufv(1− e) ≡ g (mod R(u− uvu)R).
Then
u(x+ fv(1− e))(1− eufv(1− e))u = (ux+ ufv(1− e))(1− eufv(1− e))u =
= (e+ ufv(1− e))(1− eufv(1− e))u = (e(1− eufv(1− e)) + ufv(1− e))u =
= (e+ (1− e)ufv(1− e))u = (e+ g)u ≡ u(modR(u− uvu)R).
Let p = x+ fv(1− e) and q = (1− eufv(1− e))u = (1 + eufv(1− e))−1u. Then qpq = q. Since
R−1R−1∞ = R−1∞ and R−1∞ R−1 = R−1∞ , we have q ∈ R−1q . Hence q̄p̄q̄ = q̄ in R/R(u−uvu)R. Since
q̄ ∈ (R/R(u− uvu)R)−1∞ , there exist ā, b̄ ∈ R/R(u− uvu)R such that (1̄− q̄ā)\(1̄− b̄q̄). It follows
from (1̄ − q̄p̄) = (1̄ − q̄p̄)(1̄ − q̄ā) and (1̄ − p̄q̄) = (1̄ − b̄q̄)(1̄ − b̄p̄) that (1̄ − q̄p̄)\(1̄ − p̄q̄). Then
p̄ ∈ (R/R(u− uvu)R)−1∞ . By Lemma 2.5 [6], p ∈ R−1∞ . Hence x = xux = xu(x+ fu−1q (1− e)) =
= xup.
(2) ⇒ (1). Suppose that x = xyx = xyu where u ∈ R−1∞ . Let z = yxy. Then x = xzx = xzu
and z = zxz. Hence z = z(x+(1−xz)u)z where x+(1−xz)u = u ∈ R−1∞ . z is pseudo-unit regular.
It follows from (1)⇒ (2) that there exists a p ∈ R−1∞ such that z = zuz = zup. Let e = 1− zx and
f = zu. Then e2 = e and f2 = f . It is easily checked that
fpx(1− f) + e(1− f) = 1− f and (1− f)e(1− f) = 1− f.
Then
z + e(1− f)p = fp+ e(1− f)p = (1 + fwx(1− f))−1p ∈ R−1∞ .
It is clear that x = x(z+ e(1−f)p)x with z+ e(1−f)p ∈ R−1∞ . Therefore, x is pseudo-unit regular.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
424 XIAOQING SUN, SHANGPING WANG, XIAOQIN SHEN, JIANGHUA LI
(2) ⇒ (3). It is trivial.
(3) ⇒ (2). Let x = xyx = xyw where w is quasi-unit regular. It follows from (1) ⇒ (2), we
have w = ep where e2 = e and p ∈ R−1∞ . It follows from the equation xy + (1− xy) = 1 we obtain
xyw+ (1−xy)w = w. Since x = xyw, we have x+ (1−xy)w = w. Then xy+ (1−xy)wy = wy.
Hence wy + (1− xy)(1−wy) = 1. It follows that ewy(1− e) + (1− xy)(1−wy)(1− e) = 1− e.
Consequently,
e+ (1− xy)(1− wy)(1− e) = 1− ewy(1− e) = (1 + ewy(1− e))−1
is invertible in R. Let
u = w + (1− xy)(1− wy)(1− e)p = (e+ (1− xy)(1− wy)(1− e))p.
Since R−1R−1∞ = R−1∞ and R−1∞ R−1 = R−1∞ , we have u ∈ R−1∞ . It is easy to check that
x = xyx = xyw = xyu where u ∈ R−1∞ .
Similarly, we can prove equivalences of (1), (2′), (3′).
Lemme 3.1 is proved.
Corollary 3.1. Let R be a ring and x ∈ R be regular. Then the following are equivalent:
(1) x is pseudo-unit regular;
(2) there exist some idempotent e ∈ R and some u ∈ R−1∞ such that x = eu;
(2′) there exist some idempotent e ∈ R and some u ∈ R−1∞ such that x = ue;
(3) there exist some idempotent e ∈ R and some pseudo-unit regular element w ∈ R such that
x = ew;
(3′) there exist some idempotent e ∈ R and some pseudo-unit regular element w ∈ R such that
x = we.
Proof. (1) ⇒ (2). It follows from (1)⇒ (2) of Lemma 3.1.
(2)⇒ (3). It is obvious.
(3) ⇒ (1). Assume x = xyx = ew, where e ∈ R is an idempotent and w is pseudo-unit regular.
Let w = wuw where u ∈ R−1∞ . Since xy + (1− xy) = 1, we have ewy + (1− xy) = 1. It follows
that ewy(1− e) + (1− xy)(1− e) = 1− e. Then
v := e+ (1− xy)(1− e) = 1− ewy(1− e) = (1 + ewy(1− e))−1
is a unit in R. Let
p = x+ (1− xy)(1− e)w = (e+ (1− xy)(1− e))w = vw = vwuw = vw(uv−1)vw.
Since R−1R−1∞ = R−1∞ and R−1∞ R−1 = R−1∞ , we have uv−1 ∈ R−1∞ . Then q is pseudo-unit regular.
It is easy to check that x = xyx = xy(x + (1 − xy)(1 − e)w) = xyp. The result follows from
Lemma 3.1.
Similarly, we can prove equivalences of (1), (2′), (3′).
Corollary 3.1 is proved.
By the result of Theorem 2.1 [7], an exchange ring R is a QB∞-ring if and only if every regular
element in R is pseudo-unit regular. It follows from Lemma 3.1 and Corollary 3.1, we immediately
have the following characterizations of exchange QB∞-ring.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
QUASI-UNIT REGULARITY AND QB-RINGS 425
Theorem 3.1. Let R be an exchange ring. Then the following are equivalent:
(1) R is a QB∞-ring;
(2) whenever x ∈ R is regular, there exists a u ∈ R−1∞ such that x = xyx = xyu for some
y ∈ R;
(2′) whenever x ∈ R is regular, there exists a u ∈ R−1∞ such that x = xyx = uyx for some
y ∈ R;
(3) whenever x ∈ R is regular, there exists a pseudo-unit regular element w ∈ R such that
x = xyx = xyw for some y ∈ R;
(3′) whenever x ∈ R is regular, there exists a pseudo-unit regular element w ∈ R such that
x = xyx = wyx for some y ∈ R.
By Lemma 3.1 and Theorem 3.1, the proof of Theorems 2.2, 2.3 and 2.4, Propositions 2.1, 2.2
and 2.3 could be similarly extended to QB∞-ring.
1. Ara P., Goodeal K. R., O’Meara K. C., Pardo E. Separative cancellation for projective modules over exchange rings
// Isr. J. Math. – 1998. – 105. – P. 105 – 137.
2. Ara P., Pedersen G. K., Pereva F. An infinite analogue of rings with stable range one // J. Algebra. – 2000. – 230. –
P. 608 – 655.
3. Camps R., Menal P. Power-cancellation for artinian modulea // Communs Algebra. – 1991. – 19. – P. 2081 – 2095.
4. Canfell M. J. Completions of diagrams by automorphism and Bass’first stable range condition // J. Algebra. – 1995.
– 176. – P. 480 – 513.
5. Chen H. On exchange QB-rings // Communs Algebra. – 2003. – 31. – P. 831 – 841.
6. Chen H. On QB∞-rings // Communs Algebra. – 2006. – 34. – P. 2057 – 2068.
7. Chen H. On exchange QB∞-rings // Alg. colloq. – 2007. – 14. – P. 613 – 623.
8. Warfield R. B. (Jr.) Exchange rings and decompositions of modules // Math. Ann. – 1972. – 199. – P. 31 – 36.
9. Wei J. Unit-regularity and stable range conditions // Communs Algebra. – 2005. – 33. – P. 1937 – 1946.
Received 20.07.11,
after revision — 10.02.12
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
|