On rings with weakly prime centers
We introduce a class of rings obtained as a generalization of rings with prime centers. A ring R is called weakly prime center (or simply WPC) if ab∈Z(R) (R) implies that aRb is an ideal of R where Z(R) stands for the center of R. The structure and properties of these rings are studied and the relat...
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irk-123456789-1661262020-02-20T01:27:11Z On rings with weakly prime centers Junchao Wei Статті We introduce a class of rings obtained as a generalization of rings with prime centers. A ring R is called weakly prime center (or simply WPC) if ab∈Z(R) (R) implies that aRb is an ideal of R where Z(R) stands for the center of R. The structure and properties of these rings are studied and the relationships between prime center rings, strongly regular rings, and WPC rings are discussed, parallel with the relationship between the WPC and commutativity. Введено клас кілєць, що є узагальненням кілєць з простими центрами. Кільце R називається слабко простим центром (чи просто WPC), якщо з включення ab∈Z(R) випливає, що aRb є ідеалом R, де Z(R) — центр R. Вивчено структуру i властивості таких кілець та проаналізовано співвідношення між простими центральними кільцями, сильно регулярними кільцями та кільцями з слабко простим центром паралельно зі співвідношенням між слабко простим центром та комутативністю. 2014 Article On rings with weakly prime centers / Junchao Wei // Український математичний журнал. — 2014. — Т. 66, № 12. — С. 1615–1622. — Бібліогр.: 19 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166126 512.5 en Український математичний журнал Інститут математики НАН України |
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We introduce a class of rings obtained as a generalization of rings with prime centers. A ring R is called weakly prime center (or simply WPC) if ab∈Z(R) (R) implies that aRb is an ideal of R where Z(R) stands for the center of R. The structure and properties of these rings are studied and the relationships between prime center rings, strongly regular rings, and WPC rings are discussed, parallel with the relationship between the WPC and commutativity. |
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Junchao Wei |
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Junchao Wei |
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Junchao Wei |
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On rings with weakly prime centers |
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On rings with weakly prime centers |
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On rings with weakly prime centers |
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On rings with weakly prime centers |
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On rings with weakly prime centers |
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on rings with weakly prime centers |
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Інститут математики НАН України |
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2014 |
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Статті |
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http://dspace.nbuv.gov.ua/handle/123456789/166126 |
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On rings with weakly prime centers / Junchao Wei // Український математичний журнал. — 2014. — Т. 66, № 12. — С. 1615–1622. — Бібліогр.: 19 назв. — англ. |
| series |
Український математичний журнал |
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AT junchaowei onringswithweaklyprimecenters |
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2025-07-14T20:47:31Z |
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2025-07-14T20:47:31Z |
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1837656758861955072 |
| fulltext |
UDC 512.5
Junchao Wei (School Math., Yangzhou Univ., China)
ON RINGS WITH WEAKLY PRIME CENTERS*
ПРО КIЛЬЦЯ З СЛАБКИМИ ПРОСТИМИ ЦЕНТРАМИ
We introduce a class of rings obtained as a generalization of rings with prime centers. A ring R is called weakly prime
center (or, briefly, WPC) if ab ∈ Z(R) implies that aRb is an ideal of R, where Z(R) stands for the center of R. The
structure and properties of these rings are studied, the relationships between prime center rings, strongly regular rings, and
WPC rings are discussed, parallel with the relationship between WPC to commutativity.
Введено клас кiлець, що є узагальненням кiлець з простими центрами. Кiльце R називається слабко простим
центром (чи просто WPC), якщо з включення ab ∈ Z(R) випливає, що aRb є iдеалом R, де Z(R) — центр
R. Вивчено структуру i властивостi таких кiлець та проаналiзовано спiввiдношення мiж простими центральними
кiльцями, сильно регулярними кiльцями та кiльцями з слабко простим центром паралельно зi спiввiдношенням мiж
слабко простим центром та комутативнiстю.
1. Introduction. Throughout this article, all rings considered are associative with identity, and
all modules are unital, the symbols J(R), N(R), U(R), E(R), Z(R) and Maxl(R) will stand
respectively for the Jacobson radical, the set of all nilpotent elements, the set of all invertible
elements, the set of all idempotent elements, the center and the set of all maximal left ideals of R. For
any nonempty subset X of a ring R, r(X) = rR(X) and l(X) = lR(X) denote the right annihilator
of X and the left annihilator of X, respectively.
A ring R is called
(1) reduced if N(R) = 0;
(2) Abel if E(R) ⊆ Z(R);
(3) left quasiduo if every maximal left ideal of R is an ideal;
(4) MELT if every essential maximal left ideal of R is an ideal.
Recall that a ring R has prime (semiprime ) center [8] if ab ∈ Z(R) implies a ∈ Z(R) or
b ∈ Z(R)
(
an ∈ Z(R) implies a ∈ Z(R)
)
. Clearly, commutative rings have prime center. In [8],
some basic properties of prime center rings are studied.
A ring R is called periodic [3] if for each x ∈ R, there exist distinct positive integers m and n
for which xm = xn. In [8] (Theorem 1), it is shown that for a periodic ring R, R is commutative if
and only if R has prime center.
In this paper, a new class of rings is introduced, which is a proper generalization of rings with
prime centers. A ring R is called weakly prime center (or, briefly, WPC) if ab ∈ Z(R) implies aRb
is an ideal of R. Remark 2.1 points out that WPC rings are proper generalization of rings with prime
centers. Proposition 2.6 shows that strongly regular rings are a class of WPC rings. Proposition 2.8
shows that a ring R is a division ring if and only if R is a WPC primitive ring.
Let R be a ring and e ∈ E(R). e is called left minimal idempotent if Re is a minimal left ideal
of R. We write MEl(R) for the set of all left minimal idempotents of R. A ring R is called left
min-Abel if (1 − e)Re = 0 for each e ∈ MEl(R). In [13] (Theorem 1.2), it is shown that a ring R
* This work is supported by the Foundation of Natural Science of China (11471282, 11171291) and Natural Science
Fund for Colleges and Universities in Jiangsu Province(11KJB110019).
c© JUNCHAO WEI, 2014
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 12 1615
1616 JUNCHAO WEI
is a left quasiduo ring if and only if R is a left min-Abel MELT ring. The study of left min-Abel
rings appears in [13, 15, 16]. Proposition 2.5 shows that WPC rings are left min-Abel.
Following [10], an element a of a ring R is called clean if a is a sum of a unit and an idempotent
of R, and a is said to be exchange if there exists e ∈ E(R) such that e ∈ aR and 1− e ∈ (1− a)R.
A ring R is called clean if every element of R is clean, and R is said to be exchange if every element
of R is exchange. According to [10], clean rings are always exchange, but the converse is not true,
in general. In [18], it is shown that left quasiduo exchange rings are clean; in [19], it is shown that
Abel exchange rings are clean; in [15], it is shown that quasinormal exchange rings are clean; in [16],
it is shown that weakly normal exchange rings are clean. Theorem 3.1 shows that WPC exchange
rings are clean and have stable range 1.
Following [4], a ring R is said to be semiperiodic if for each x ∈ R\(J(R) ∪ Z(R)), there
exist m,n ∈ Z, of opposite parity, such that xn − xm ∈ N(R). Clearly, the class of semiperiodic
rings contains all commutative rings, all Jacobson radical rings, and certain non-nil periodic rings.
Theorem 4.2 shows that for a semiperiodic ring R with J(R) 6= N(R), R is WPC if and only if R
is commutative.
2. Some properties of WPC rings.
Definition 2.1. A ring R is called weakly prime center (WPC) if for any a, b ∈ R, ab ∈ Z(R)
implies aRb is an ideal of R.
Clearly commutative rings are WPC.
Proposition 2.1. Prime center rings are WPC.
Proof. Let R be a prime center ring and a, b ∈ R with ab ∈ Z(R). Since R is prime center,
a ∈ Z(R) or b ∈ Z(R), one has aRb = abR = Rab. Hence aRb is an ideal of R and R is WPC.
Recall that a ring R is directly finite if ab = 1 implies ba = 1 for any a, b ∈ R. In [8], it is shown
that prime center rings are directly finite.
Lemma 2.1. WPC rings are directly finite.
Proof. Let a, b ∈ R with ab = 1. Let e = ba. Then a = ae and e ∈ E(R). Since a(1 − e) =
= 0 ∈ Z(R), aR(1− e) is an ideal of R, that is, R(1− e) is an ideal of R because aR = R. Hence
(1 − e)a ∈ R(1 − e), which implies (1 − e)a = (1 − e)ae = 0. Then a = ea and 1 = ab = eab =
= e = ba, this shows that R is a directly finite ring.
Proposition 2.2. Let R be a local ring. If J(R) is commutative, then R is WPC.
Proof. Assume that ab ∈ Z(R). Then Rab is an ideal of R. If Rab = R, then ab ∈ U(R),
by Lemma 2.1, a, b ∈ U(R), so aRb = R is an ideal of R. If Rab ⊆ J(R), then ab /∈ U(R).
If a /∈ U(R) and b /∈ U(R), then aR,Rb ⊆ J(R). Since J(R) is commutative, RaRbR =
= R
(
a(Rb)
)
R = RbaR = aRb, this gives aRb is an ideal of R. If a ∈ U(R) and b /∈ U(R),
then RaRbR = RbR = RabR = Rab = Rb = aRb, so aRb is an ideal. Similarly, if a /∈ U(R) and
b ∈ U(R), we can show that aRb is an ideal. Hence R is WPC.
Proposition 2.3. If R be a local prime center ring, then R is commutative.
Proof. It is an immediate result of [8] (Basic Lemma 2(b)).
Remark 2.1. By Proposition 2.2, one knows that division rings are WPC. By Proposition 2.3,
noncommutative division rings need not be prime center. Thus there exists a WPC ring (noncom-
mutative division rings) which is not prime center. Hence WPC rings are proper generalization of
prime center rings.
Proposition 2.4. If R is a semiprime WPC ring, then R is reduced.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 12
ON RINGS WITH WEAKLY PRIME CENTERS 1617
Proof. If N(R) 6= 0, then there exists 0 6= a ∈ N(R) such that a2 = 0. Since R is WPC, aRa
is an ideal of R, this leads to aRaR ⊆ aRa, so aRaRa ⊆ aRa2 = 0. Since R is semiprime, a = 0,
which is a contradiction. Thus N(R) = 0.
Recall that a ring R is NCI [7] if either N(R) = 0 or N(R) contains a nonzero ideal of R. By
the proof of Proposition 2.4, we have the following corollary.
Corollary 2.1. WPC rings are NCI.
Remark 2.2. [7], Example 2.5, points out that NCI rings need not be directly finite, by
Lemma 2.1, we know that the converse of Corollary 2.1 is not true.
Remark 2.3. Simple rings need not be WPC. For example, let D be a division ring and
R =
(
D D
D D
)
. Then R be a simple ring. Clearly,
(
1 0
0 0
)(
0 0
0 1
)
= 0 ∈ Z(R) and(
1 0
0 0
)
R
(
0 0
0 1
)
=
(
0 D
0 0
)
. If R is WPC, then(
0 1
0 1
)
=
(
1 1
1 1
)(
1 0
0 0
)(
0 1
0 0
)(
0 0
0 1
)
∈
(
1 0
0 0
)
R
(
0 0
0 1
)
,
which is a contradiction. Hence R is not a WPC ring.
Recall that a ring R is left min-Abel [13] if for every e ∈ MEl(R) =
{
e ∈ E(R)| Re is a
minimal left ideal of R
}
, e is left semicentral in R. Clearly, R is a left min-Abel ring if and only if
(1− e)Re = 0 for each e ∈MEl(R).
Proposition 2.5. If R is a WPC ring, then R is left min-Аbel.
Proof. Let e ∈ MEl(R). Since R is a WPC ring and (1 − e)e = 0 ∈ Z(R), (1 − e)Re is
an ideal of R, this gives R(1 − e)Re ⊆ (1 − e)Re. If (1 − e)Re 6= 0, then R(1 − e)Re = Re, so
e ∈ eRe = eR(1− e)Re ⊆ e(1− e)Re = 0, which is a contradiction. Therefore (1− e)Re = 0 and
R is a left min-Abel ring.
Remark 2.4. The converse of Proposition 2.5 is not true in general. For example, let
R =
(
Z5 Z5
0 Z5
)
. Cleary,
(
2 4
0 4
)(
4 1
0 2
)
=
(
3 0
0 3
)
∈ Z(R) and
(
2 4
0 4
)
R
(
4 1
0 2
)
=
=
{(
4 1
0 2
)
|x, y ∈ Z5
}
. But
(
3 0
0 3
)(
4 1
0 0
)
=
(
2 3
0 0
)
/∈
(
2 4
0 5
)
R
(
4 1
0 2
)
, thus R is
not WPC. Since R is a left quasiduo ring by [18], R is a left min-Abel ring by [13] (Theorem 1.2).
Recall that a ring R is von Neumann regular if a ∈ aRa for any a ∈ R, and R is said to be
strongly regular if a ∈ a2R for any a ∈ R. It is well known that a ring R is a strongly regular ring
if and only if R is a reduced von Neumann regular ring.
Proposition 2.6. The following conditions are equivalent for a ring R :
(1) R is a strongly regular ring;
(2) R is a WPC von Neumann regular ring.
Proof. (1) =⇒ (2). Since R is a strongly regular ring, R is an Abel von Neumann regular ring.
Hence, for any a, b ∈ R, aR = eR = Re for some e ∈ E(R), this gives aRb = R(eb) = Rg = gR
for some g ∈ E(R). Thus aRb is an ideal of R and R is a WPC ring.
(2) =⇒ (1). Since von Neumann regular rings are semiprime, by Proposition 2.4, R is reduced.
Hence R is a strongly regular ring.
Recall that a ring R is left SF if every simple left R-module is flat. It is well known that von
Neumann regular rings are left SF. In [11] (Remark 3.13), it is shown that if R is a reduced left SF
ring, then R is strongly regular. We can generalize this result as follows.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 12
1618 JUNCHAO WEI
Proposition 2.7. If R is a prime center left SF ring, then R is a commutative strongly regular
ring.
Proof. Let a ∈ R with a2 = 0. By [8] (Basic Lemma 2(a)), a ∈ Z(R). If a 6= 0, then l(a) 6= R
and there exists a maximal left ideal M of R such that l(a) ⊆ M. Since R is a left SF ring, R/M
is flat as left R-module. Since a ∈ l(a) ⊆M, a = am for some m ∈M. Since a ∈ Z(R), a = ma,
one obtains 1 − m ∈ l(a) ⊆ M, 1 ∈ M, which is a contradiction. Hence a = 0, which implies
R is reduced, by [11] (Remark 3.13), R is strongly regular. Now let x ∈ R. Then x = xyx for
some y ∈ R. Write e = xy and g = yx. Then e, g ∈ E(R) and x = ex = xg. Since R is Abel,
e, g ∈ Z(R). Since yx = g ∈ Z(R), y ∈ Z(R) or x ∈ Z(R). If x ∈ Z(R), we are done. If
y ∈ Z(R), then for any r ∈ R, we have xr = xgr = xrg = xryx = xyrx = erx = rex = rx,
which implies x ∈ Z(R). Hence R is commutative.
Corollary 2.2. If R is a prime center von Neumann regular ring, then R is a commutative
strongly regular ring.
Remark 2.5. Since strongly regular rings need not be commutative, by Corollary 2.2, strongly
regular rings need not be prime center.
Corollary 2.3. R is a field if and only if R is a prime center division ring.
Proof. Fields are certainly prime center division rings. The converse is an immediate corollary
of Corollary 2.2.
Proposition 2.8. R is a division ring if and only if R is a WPC primitive ring.
Proof. Division rings are certainly WPC primitive rings. Now let R be a WPC primitive ring.
If R is not a division ring, then there exists a subring S of R such that S ∼=
(
D D
D D
)
, where D is a
division ring. Clearly,
(
D D
D D
)
is not reduced, so S is not reduced, this implies R is not reduced.
But by Proposition 2.4, R is reduced, which is a contradiction. Hence R is a division ring.
Corollary 2.3 and Proposition 2.8 give the following corollary.
Corollary 2.4. R is a field if and only if R is a prime center primitive ring.
A ring R is called weakly regular if a ∈ aRaR∩RaRa for every a ∈ R. A left R-module M is
called YJ-injective (Wnil-injective (see [14])) if for each 0 6= a ∈ R
(
0 6= a ∈ N(R)
)
, there exists a
positive integer n such that an 6= 0 and each left R-homomorphism Ran −→M can be extended to
R −→M. It is easy to see that Y J-injective modules are Wnil-injective.
Proposition 2.9. Let R be a WPC ring. If each singular simple left R-modules are Wnil-
injective, then R is reduced.
Proof. By Proposition 2.4, we only need to show that R is semiprime. Assume that a ∈ R with
aRa = 0. If a 6= 0, then there exists a maximal left idealM of R such that r(aR) ⊆M.We claim that
M is an essential left ideal of R. If not, M = l(e) for some e ∈MEl(R). Since R is a WPC ring,
R is left min-Abel by Proposition 2.5. Hence aRe = aeRe = 0 because a ∈ r(aR) ⊆ M = l(e),
this leads to e ∈ r(aR) ⊆ l(e), which is a contradiction. Thus M is an essential left ideal of R
and R/M is a singular simple left R-module, by hypothesis, R/M is Wnil-injective. Then the left
R-homomorphism f : Ra −→ R/M defined by f(ra) = r +M can be extended into R −→ R/M,
so there exists d ∈ R such that 1− ad ∈M. Since ad(ad) = 0, 1− ad is a unit of R, so M = R, a
contradiction. Hence a = 0.
Corollary 2.5. Let R be a WPC ring whose singular simple left R-modules are Y J-injective,
then R is a reduced weakly regular ring.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 12
ON RINGS WITH WEAKLY PRIME CENTERS 1619
Proof. By Proposition 2.9, R is reduced. By [9] (Theorem 4), R is a reduced weakly regular
ring.
3. Exchange WPC rings. Following [10], an element a of a ring R is called clean if a is a
sum of a unit and an idempotent of R, and a is said to be exchange if there exists e ∈ E(R) such
that e ∈ aR and 1− e ∈ (1− a)R. A ring R is called clean if every element of R is clean, and R
is said to be exchange if every element of R is exchange. According to [10], clean rings are always
exchange, but the converse is not true unless R satisfies one of the following conditions: (1) R is a
left quasiduo ring [18]; (2) R is an Abel ring [19]; (3) R is a quasinormal ring [15]; (4) R is a weakly
normal ring [16].
Theorem 3.1. Let R be a WPC ring and a ∈ R. Then
(1) If a is exchange, then a is clean.
(2) If R is an exchange ring, then R is a clean ring.
(3) If an is clean for some n ≥ 1, then a is clean.
(4) If a2 is clean, then a and −a are clean.
Proof. (1) Let e ∈ E(R) such that e ∈ aR and 1−e ∈ (1−a)R.Write e = ab and 1−e = (1−a)c
for some b = be, c = c(1 − e) ∈ R. Then (a − (1 − e))(b − c) = ab − ac − (1 − e)b + (1 − e)c =
= ab + (1 − a)c − (1 − e)b − ec = 1 − (1 − e)b − ec. Since R is a WPC ring and b(1 − e) =
= 0 ∈ Z(R), bR(1 − e) is an ideal of R. Hence bR(1 − e)R ⊆ bR(1 − e) and bR(1 − e)Re = 0,
which implies bR(1 − e)Rb = bR(1 − e)Rbe = 0. Therefore (R(1 − e)Rb)2 = 0, this leads to
(1 − e)b ∈ R(1 − e)Rb ⊆ J(R). Similarly, ec ∈ J(R). Hence 1 − (1 − e)b − ec is a unit of R, by
Lemma 2.1, one obtains a− (1− e) is an unit of R. Hence a is a clean element.
(2) It is an immediate result of (1).
(3) Since an is clean, there exist u ∈ U(R) and f ∈ E(R) such that an = u+ f. Let e = u(1−
−f)u−1. Then (an−e)u = (u+f)u−u(1−f) = an(an−1) ∈ aR, so e = an+(an−a2n)u−1 ∈ aR
and 1− e ∈ (1− a)R, this implies a is exchange, by (1), a is clean.
(4) Since a2 = (−1a)2 is clean, by (3), a and −a are clean.
Corollary 3.1. Let R be a WPC ring and idempotent can be lifted modulo J(R). Let a ∈ R be
clean and e ∈ E(R). Then
(1) ae is clean.
(2) If −a is also clean, then a+ e is clean.
Proof. (1) Since a is clean, ā is clean in R̄ = R/J(R). Since R is a WPC ring, eR(1−e) is an
ideal of R, which implies ((1− e)ReR)2 = (eR(1− e)R)2 = 0. Hence (1̄− ē)R̄ē = ēR̄(1̄− ē) = 0̄,
that is, ē is a central idempotent in R̄. Since a is clean in R, there exist u ∈ U(R) and f ∈ E(R)
such that a = u+f. Let v ∈ R such that uv = vu = 1. Then, in R̄, āē = (ūē+ ē− 1̄)+(f̄ ē+1̄− ē).
Clearly, (ūē+ ē− 1̄)(v̄ē+ ē− 1̄) = (v̄ē+ ē− 1̄)(ūē+ ē− 1̄) = 1̄ and (f̄ ē+ 1̄− ē)2 = f̄ ē+ 1̄− ē,
so āē is clean in R̄. Since idempotent can be lifted modulo J(R), there exists g ∈ E(R) such that
ḡ = f̄ ē+ 1̄− ē. Let w ∈ R such that w̄ = ūē+ ē− 1̄. Then w ∈ U(R) and ae−w− g ∈ J(R). Let
ae − w − g = x ∈ J(R). Then ae = g + w(1 + w−1x). Since w(1 + w−1x) ∈ U(R), ae is clean
in R.
(2) Since−a is clean inR, 1+a is clean inR.Hence ā and 1̄+ā are all clean in R̄ = R/J(R). Let
ā = ū+f̄ and 1̄+ā = v̄+ ḡ where u, v ∈ U(R) and f, g ∈ E(R). Clearly, ā+ ē = ā(1̄− ē)+(1̄+ā)ē,
so ā + ē = v̄ē + ū(1̄ − ē) + ḡē + f̄(1̄ − ē). Clearly, (v̄ē + ū(1̄ − ē))(v̄−1ē + ū−1(1̄ − ē)) = 1̄ and
ḡē+ f̄(1̄− ē) ∈ E(R̄). Therefore, ā+ ē is clean in R̄, similar to (1), we obtain a+ e is clean in R.
In [5], it is showed that if R is a unit regular ring, then every element of R is a sum of two units.
A ring R is called an (S, 2)-ring [5], if every element of R is a sum of two units of R. In [2], it
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1620 JUNCHAO WEI
is proved that if R is an Abel π-regular ring, then R is an (S, 2)-ring if and only if Z/2Z is not a
homomorphic image of R.
Theorem 3.2. Let R be a WPC π-regular ring. Then R is an (S, 2)-ring if and only if Z/2Z
is not a homomorphic image of R.
Proof. Since R is a WPC π-regular ring, R/J(R) is π-regular ring. Since R is an exchange
ring, idempotent can be lifted modulo J(R). By the proof of Corollary 3.1(1), R/J(R) is an Abel
ring. By [2], R/J(R) is an (S, 2)-ring if and only if Z/2Z is not a homomorphic image of R/J(R).
By [15] (Lemma 4.3), we are done.
In light of Theorem 3.2, we have the following corollaries:
Corollary 3.2. Let R be a WPC π-regular ring such that 2 = 1 + 1 ∈ U(R). Then R is an
(S, 2)-ring.
Corollary 3.3. Let R be a WPC π-regular ring. Then R is an (S, 2)-ring if and only if for
some d ∈ U(R), 1 + d ∈ U(R).
Recall that a ring R is said to have stable range 1 [12] if for any a, b ∈ R satisfying aR+bR = R,
there exists y ∈ R such that a + by is right invertible. Clearly, R has stable range 1 if and only if
R/J(R) has stable range 1. In [19] (Theorem 6), it is showed that exchange rings with all idempotents
central have stable range 1.
Theorem 3.3. WPC exchange rings have stable range 1.
Proof. Let R be a WPC exchange ring. Then R/J(R) is exchange with all idempotents central,
so, by [19] (Theorem 6), R/J(R) has stable range 1. Therefore R has stable range 1.
In [17], A ringR is said to satisfy the unit 1-stable condition if for any a, b, c ∈ R with ab+c = 1,
there exists u ∈ U(R) such that au+ c ∈ U(R). It is easy to prove that R satisfies the unit 1-stable
condition if and only if R/J(R) satisfies the unit 1-stable condition.
Theorem 3.4. Let R be a WPC exchange ring, then the following conditions are equivalent:
(1) R is an (S, 2)-ring.
(2) R satisfies the unit 1−stable condition.
(3) Every factor ring of R is an (S, 2)-ring.
(4) Z2 is not a homomorphic image of R.
A ring R is called left topologically boolean, or a tb-ring [1] for short, if for every pair of distinct
maximal left ideals of R there is an idempotent in exactly one of them.
Theorem 3.5. Let R be a WPC exchange ring. Then R is a left tb-ring.
Proof. Suppose that M and N are distinct maximal left ideals of R. Let a ∈ M\N. Then
Ra + N = R and 1 − xa ∈ N for some x ∈ R. Clearly, xa ∈ M\N. Since R is a WPC
exchange ring, R is clean by Theorem 3.1, there exist an idempotent e ∈ E(R) and a unit u in R
such that xa = e + u. If e ∈ M, then u = xa − e ∈ M from which it follows that R = M, a
contradiction. Thus e /∈ M. If e /∈ N, then Re + N = R. Since R is a WPC ring, by the proof of
Corollary 3.1(1), (1− e)ReR ⊆ J(R) ⊆ N, 1− e ∈ (1− e)R = (1− e)Re+ (1− e)N ⊆ N. Hence
u = (1− e) + (xa− 1) ∈ N. It follows that N = R which is also impossible. We thus have that e
is an idempotent belonging to N only.
4. WPC semiperiodic rings. Following [4], a ring R is said to be semiperiodic if for each
x ∈ R\(J(R)∪Z(R)), there exist m,n ∈ Z, of opposite parity, such that xn−xm ∈ N(R). Clearly,
the class of semiperiodic rings contains all commutative rings, all Jacobson radical rings, and certain
nonnil periodic rings.
Lemma 4.1. If R is a WPC semiperiodic ring, then N(R) ⊆ J(R).
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ON RINGS WITH WEAKLY PRIME CENTERS 1621
Proof. Let a ∈ N(R) with ak = 0, and let x ∈ R. If ax ∈ J(R), then ax is right quasiregular;
and if ax ∈ Z(R), then ax is nilpotent and again ax is right quasiregular. Suppose, then, that
ax /∈ J ∪ Z, in which case [4] (Lemma 2.3 (iii)) gives q ∈ Z+ and an idempotent e of form ay such
that (ax)q = (ax)qe. Since
e = ay = eay = ea(1− e)y + eaey = ea(1− e)y + ea2y2 =
= ea(1− e)y + ea2(1− e)y2 + ea2ey2 = ea(1− e)y + ea2(1− e)y2 + ea3y3 = . . .
. . . =
k−1∑
i=1
eai(1− e)yi + eakyk =
k−1∑
i=1
eai(1− e)yi.
Since R is aWPC ring, eR(1−e) ∈ J(R) by the proof of Corollary 3.1(1), which implies e ∈ J(R),
so e = 0 and (ax)q = 0, which shows that ax is right quasiregular. Thus a ∈ J(R).
Theorem 4.1. If R is a WPC semiperiodic ring, then R/J(R) is commutative.
Proof. Let R̄ = R/J(R). Clearly, N(R) ⊆ J(R) by Lemma 4.1. Now let ā ∈ R̄ with ā2 = 0.
Then a2 ∈ J(R) ⊆ N(R) ∪ Z(R) by [4] (Lemma 2.6). If a2 ∈ N(R), then a ∈ N(R). Hence
a ∈ J(R) by Lemma 4.1 and ā = 0̄. If a2 ∈ Z(R), then ā2 ∈ Z(R̄). If ā ∈ Z(R̄), then āR̄ā = 0̄.
Since R̄ is semiprime, ā = 0̄. If ā /∈ Z(R̄), then a /∈ J(R) ∪ Z(R). By [4] (Lemma 2.3(iii)),
aq = aqe for some q ≥ 1 and e ∈ E(R) with the form ay. Hence e = eay = ea(1− e)y + eaey =
= ea(1 − e)y + ea2y2 ∈ J(R). Thus e = 0 and aq = 0. This implies a ∈ N(R) ⊆ J(R) by
Lemma 4.1, which is a contradiction. Hence ā ∈ Z(R̄) and so ā = 0̄. Therefore R̄ is a reduced ring.
Since R̄ is also semiperiodic, by [4] (Lemma 4.4), R̄ is commutative.
Theorem 4.2. Let R be a WPC semiperiodic ring. Then
(1) N(R) is an ideal of R.
(2) If J(R) 6= N(R), then R is commutative.
Proof. (1) Let a, b ∈ N(R) and x ∈ R. Then a − b, ax ∈ J(R) by Lemma 4.1. By [4]
(Lemma 2.6), a − b, ax ∈ N(R) ∪ Z(R). If a − b, ax ∈ N(R), we are done. If a − b, ax ∈ Z(R).
Then (a−b)a = a(a−b) and (ax)n = anxn for any n ≥ 1, this gives ab = ba, thus a−b, ax ∈ N(R).
Similarly, xa ∈ N(R). Therefore N(R) is an ideal of R.
(2) By [4] (Lemma 2.6), it follows that
J(R) = (J(R) ∩N(R)) ∪ (J(R) ∩ Z(R). (4.1)
By (1), viewing (4.1) as a relation holding on additive subgroup, we conclude that
J(R) = J(R) ∩N(R) or J(R) = J(R) ∩ Z(R).
This implies that
J(R) ⊆ N(R) or J(R) ⊆ Z(R).
Since J(R) 6= N(R), by Lemma 4.1, J(R) ⊆ Z(R).
Now let x ∈ R. If x /∈ Z(R), then x /∈ J(R) ∪ Z(R), so there exists positive integers n, m
(n ≥ m) of opposite parity such that xn − xm ∈ N(R). Let k ≥ 1 such that (xn − xm)k = 0. Then
((x − xn−m+1)m)k = 0, this gives x − xn−m+1 ∈ N(R) ⊆ J(R) ⊆ Z(R). By Herstein’s theorem
[6], R is commutative.
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1622 JUNCHAO WEI
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Received 15.10.12
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