Polynomial extensions of generalized quasi-Baer rings
In this paper, we consider the behavior of polynomial rings over generalized quasi-Baer rings and show that the generalized quasi-Baer condition on a ring R is preserved by many polynomial extensions.
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Institute of Mathematics, NAS of Ukraine
2010
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508893376937984 |
|---|---|
| author | Ghalandarzadeh, S. Javadi, H. S. Khoramdel, M. Галандарзадех, С. Яваді, Г. С. Хорамдель, М. |
| author_facet | Ghalandarzadeh, S. Javadi, H. S. Khoramdel, M. Галандарзадех, С. Яваді, Г. С. Хорамдель, М. |
| author_sort | Ghalandarzadeh, S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:39:51Z |
| description | In this paper, we consider the behavior of polynomial rings over generalized quasi-Baer rings and show that the generalized quasi-Baer condition on a ring R is preserved by many polynomial extensions. |
| first_indexed | 2026-03-24T02:32:26Z |
| format | Article |
| fulltext |
K O R O T K I P O V I D O M L E N N Q
UDC 517.5
Sh. Ghalandarzadeh (K.N. Toosi Univ. Technology,Tehran, Iran),
H. S. Javadi (Shahed Univ., Tehran, Iran),
M. Khoramdel (K.N. Toosi Univ. Technology,Tehran, Iran)
POLYNOMIAL EXTENSIONS
OF GENERALIZED QUASI-BAER RINGS
POLINOMIAL|NI ROZÍYRENNQ
UZAHAL|NENYX KVAZIBEROVYX KILEC|
In this paper we consider the behavior of polynomial rings over generalized quasi-Baer rings, and we
show that the generalized quasi-Baer condition on a ring R is preserved by many polynomial extensions.
Rozhlqnuto povedinku polinomial\nyx kilec\ nad uzahal\nenymy kvaziberovymy kil\cqmy i po-
kazano, wo uzahal\nena kvaziberova umova wodo kil\cq R zberiha[t\sq pry bahat\ox
polinomial\nyx rozßyrennqx.
1. Introduction. Throughout this paper all rings are associative with identity. A ring
R is called (quasi-)Baer if the right annihilator of every (right ideal) nonempty subset
of R is generated as a right ideal by an idempotent. It is easy to see that the Baer and
quasi-Baer properties are left-right symmetric for any ring. The study of Baer rings has
its roots in functional analysis. In [1] Kaplansky introduced Baer rings to abstract
various properties of von Neumann algebras and complete ∗-regular rings. In [2]
Clark uses the quasi-Baer concept to characterize when a finite-dimensional algebra
with unity over an algebraically closed field is isomorphic to a twisted matrix units
semigroup algebra. The concepts of Baer and quasi-Baer have been investigated by
several authors for rings. Every prime ring is a quasi-Baer ring. Since Baer rings are
nonsingular, the prime rings R with Z Rr ( ) ≠ 0 are quasi-Baer but not Baer. Ano-
ther generalization of Baer rings are p.p.-rings. A ring R is called a right (resp. left)
p.p.-ring if every principal right (resp. left) ideal is projective (equivalently, if the
right (resp. left) annihilator of any element of R is generated by an idempotent of R ).
A ring R is called a p.p.-ring if it is both right and left p.p.-ring. A ring R is said to
be generalized right p.p.-ring if for any x R∈ the right annihilator of xn is gene-
rated by an idempotent for some positive integer n. Von Neumann regular rings are
p.p.-rings, and π-regular rings are generalized p.p.-rings in the same sense as von
Neumann regular rings.
In [3, 4], Birkenmeier, Kim and Park introduced a principally quasi-Baer ring and
used them to generalize many results on reduced (i.e., it has no nonzero nilpotent
elements) p.p.-rings. A ring R is called right principally quasi-Baer (or simply
right p.q.-Baer) if the right annihilator of a principal right ideal is generated by an
idempotent. Similarly, left p.q.-Baer rings can be defined. In [5] Moussavi, Javadi
and Hashemi introduced generalized (principally) quasi-Baer ring. A ring R is
generalized right (principally) quasi-Baer if for any (principal) right ideal I of R,
the right annihilator of I n is generated by an idempotent for some positive integer n,
depending on I. For example Z
pn , n > 2 ( p is a prime number), is generalized
quasi-Baer but is not quasi-Baer.
In 1974 Armendariz seems to be the first to consider the behavior of polynomial
© SH. GHALANDARZADEH, H. S. JAVADI, M. KHORAMDEL, 2010
698 ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
POLYNOMIAL EXTENSIONS OF GENERALIZED QUASI-BAER RINGS 699
rings over Baer rings [6] (Theorem B). In this paper we consider the behavior of
polynomial rings over generalized quasi-Baer rings.
We used R x[ ] , R[ , , ]x α δ , R x x[ , ]−1 , r XR( ) , l XR( ) and Id( )R for the ring of
polynomial over R , the skew polynomial ring over R , the laurent polynomial ring
over R , the right and left annihilators of X subset of R and the set of all idempotent
of R , respectively.
2. Main results. In this section we prove our main result showing that the
generalized quasi-Baer condition on R is preserved by many polynomial extensions.
Lemma 1. Let I be an right ideal of the ring R then we have the following
assertions:
(1) I xn[ ] = ( )[ ]I x n ;
(2) r I xR x[ ]( )[ ] = r I xR( )[ ] .
Proof. The proof is straightforward.
Recall that a ring R is called Armendariz if whenever polynomials
f x( ) = a a x a xm
m
0 1+ + … + and g x( ) = b b x b x R xn
n
0 1+ + … + ∈ [ ]
satisfy f x g x( ) ( ) = 0 then a bi j = 0 for all i, j.
Let cf denote the set of all coefficients of f x R( ) ∈ .
Proposition 1. Let R be a generalized right quasi-Baer and Armendariz ring.
Then R x[ ] is a generalized right quasi-Baer ring.
Proof. Assume R be a generalized right quasi-Baer and Armendariz ring. Let I
be a right ideal of R x[ ] and I0 denote the set of coefficients of all elements of I in
R . It is clear that I0 is a right ideal of R , thus there exists e R∈ Id( ) such that
r IR
n( )0 = eR for some n N∈ . We claim that r IR x
n
[ ]( ) = eR x[ ] . It is clear that
I I x⊆ 0[ ] , then from Lemma 1 eR x[ ] = r I xR x
n
[ ]( )[ ]0 ⊆ r IR x
n
[ ]( ) . Conversely let
g x( ) = b b x b x r In
n
R x
n
0 1+ + … + ∈ [ ]( ) and a = a a a Ii i ii
k n
n1 21 0… ∈
=∑
with a Ii j
∈ 0 .
Then there exists f Ii j
∈ such that a ci fj i j
∈ . Therefore f x f xi i1 2
( ) ( )…
… f x g xin
( ) ( ) = 0, then a a a bi i i in1 2
… = 0, since R is Armendariz ring. Thus
g x( ) ∈ r I xR x
n
[ ]( )[ ]0 = eR x[ ] .
The proposition is proved.
We know that, if R be quasi-Baer ring then R x[ ] is quasi-Baer [3] (Theorem 1.2).
By Proposition 1 we showed that, if R is an Armendariz generalized right quasi-Baer
ring then R x[ ] is a generalized right quasi-Baer ring. Also in Proposition 2 we will
prove the converse of Proposition 1 is correct without Armendariz property. But in
fact, we do not know of any example of generalized quasi-Baer polynomial ring such
that R is a generalized quasi-Baer but R is not Armendariz.
Question: Let R be a generalized right quasi-Baer ring. Is R x[ ] generalized
right quasi-Baer ring without Armendariz property?
Proposition 2. Let R x[ ] be a generalized right quasi-Baer ring then R is a
generalized right quasi-Baer ring.
Proof. Let R x[ ] be generalized right quasi-Baer ring and I be a right ideal of
R . Then there exists an idempotent e x R x( ) [ ]∈ such that r I xR x
n
[ ]( )[ ] = e x R x( ) [ ]
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
700 SH. GHALANDARZADEH, H. S. JAVADI, M. KHORAMDEL
for some n N∈ . Let e0 be constant term of e x( ) then e0
2 = e0 . Since I e xn ( ) =
= 0, we have I en
0 = 0 therefore e r IR
n
0 ∈ ( ) . Thus e R r IR
n
0 ⊆ ( ) .
Conversely, let b r IR
n∈ ( ) then b r I x RR x
n∈ [ ]( )[ ] ∩ = e x R x R( ) [ ] ∩ . There-
fore we have b = e x h x( ) ( ) for some h x R x( ) [ ]∈ . Thus b = e h0 0 where h0 is
constant term of h x( ) so b e R∈ 0 . Hence r IR
n( ) = e R0 .
The proposition is proved.
Proposition 3. Let ∆ be a multiplicatively closed subset of R consisting of
central regular element. Then:
(1) If R is generalized right quasi-Baer ring then ∆−1R is generalized right
quasi-Baer ring.
(2) Let Id( )R = Id( )∆−1R . If ∆−1R is generalized right quasi-Baer then R
is generalized right quasi-Baer ring.
(3) If R x[ ] is generalized right quasi-Baer ring then R x x[ , ]−1 is generalized
right quasi-Baer ring.
(4) Let Id( )R = Id( )[ , ]x x−1 . I f R x x[ , ]−1 is generalized right quasi-Baer
then R is generalized right quasi-Baer ring.
Proof. (1) Assume that R is a generalized right quasi-Baer ring. Let I be a
right ideal of ∆−1R and I0 = { },a R b a I b∈ ∈ ∈−1 for some ∆ . It is clear I0 ≠
≠ ∅, I0 ≠ R , I0 � R and ( )∆−1
0I n = ∆−1
0I n .
We know I ⊆ ∆−1
0I . Now let c d I− −∈1 1
0∆ such that c ∈∆ , d I∈ 0 . Thus
there exists k ∈∆ such that k d I− ∈1 . Since d I∈ 0 , therefore c d−1 =
= k dc k I− − ∈1 1 hence ∆− ⊆1
0I I .
Now we claim ∆−1
0r IR( ) = r I
R∆ ∆−
−
1
1
0( ) . Let a b r IR
− −∈1 1
0∆ ( ) then
( )( )c d a b− −1 1 = 0 for all c d I− −∈1 1
0∆ , since db = 0. Thus a b r I
R
− −∈ −
1 1
01∆ ∆( ) ,
therefore ∆−1
0r IR( ) ⊆ r I
R∆ ∆−
−
1
1
0( ) .
Conversely, let a b r I
R
− −∈ −
1 1
01∆ ∆( ) then c d a b− −1 1( ) = 0, for all
c d I− −∈1 1
0∆ . Thus db = 0 then b r IR∈ ( )0 . Therefore a b r IR
− −∈1 1
0∆ ( ) .
By hypothesis, r IR
n( )0 = eR for some e2 = e R∈ . Thus I en
0 = 0 and so 0 =
= ∆−1
0I en = ( )∆−1
0I n = I en . Hence e R∆−1 ⊆ r I
R
n
∆−1 ( ) . Let a b r I
R
n− ∈ −
1
1∆ ( ) .
Then 0 = I a bn −1 = ( )∆− −1
0
1I a bn = ∆− −1
0
1I a bn and so b r I eRR
n∈ =( )0 . Hence
a b e R− −∈1 1∆ . Therefore r I
R
n
∆−1 ( ) = e R∆−1 .
(2) Let ∆−1R is generalized right quasi-Baer. We prove that R is generalized
right quasi-Baer ring. Let I be a right ideal of R then ∆−1I is right ideal of ∆−1R ,
thus there exists e R∈ such that e2 = e and r I
R
n
∆ ∆−
−
1
1(( ) ) = e R( )∆−1 for some
n N∈ . We prove r I eRR
n( ) = . We show that r I eRR
n( ) ⊆ . Let b r IR
n∈ ( ) then
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
POLYNOMIAL EXTENSIONS OF GENERALIZED QUASI-BAER RINGS 701
I bn = 0, thus 0 = ∆−1I bn = ( )∆−1I bn and so b r I e R
R
n∈ =−
− −
∆ ∆ ∆1
1 1(( ) ) ( ) . It
follows that b = eb eR∈ . The other side is similarly.
(3), (4) Let ∆ = { }, , ,1 2x x … , then ∆− −=1 1R x R x x[ ] [ , ] , and so proof is
complete.
Recall that for a ring R with a ring endomorphism α : R R→ and α-derivation
δ : R R→ , the Ore extension R x[ , , ]α δ of R is the ring obtained by giving the po-
lynomial ring over R with the new multiplication xr = α δ( ) ( )r x r+ for all r R∈ .
If δ = 0, we write R x[ , ]α for R x[ , , ]α 0 and is called an Ore extension of endo-
morphism type (also called a skew polynomial ring). In [7] Kerempa defined the rigid
rings. Let α be an endomorphism of R, α is called a rigid endomorphism if r rα( ) =
= 0 implies r = 0 for r R∈ . A ring R is called to be α-rigid if there exist a rigid
endomorphism α of R . If R be a α-rigid then Id( )R = Id( )[ , , ]R x α δ =
= Id( )[ , ]R x α (Corollary 7). Let R be a rigid ring. It is clear that generalized quasi-
Baer and quasi-Baer conditions are equivalent. Then if R be α-rigid ring, R is
generalized quasi-Baer if and only if R x[ , ]α is generalized quasi-Baer ring [8]
(Corollary 12). In Example 1 we show that rigid condition is not superfluous.
Example 1. Let Z be the ring of integers and consider the ring Z Z⊕ with the
usual addition and multiplication. Then the subring R = {( , )a b Z Z a∈ ⊕ ≡
≡ b (mod )}2 of Z Z⊕ is commutative reduced ring. Note that only idempotents of
R are (0, 0) and (1, 1). Hence from [5] (Example 2.1) R is not generalized right
quasi-Baer. Now let α : R R→ be defined by α(( , ))a b = ( , )b a Then α is an au-
tomorphism of R. Hence R x[ , ]α is quasi-Baer from [8] (Example 9).
1. Kaplansky I. Rings of operators. – New York: Benjamin, 1965.
2. Clark W. E. Twisted matrix units semigroup algebras // Duke Math. J. – 1967. – 34. – P. 417 – 424.
3. Birkenmeier G. F., Kim J. Y., Park J. K. Polynomial extensions of Baer and quasi-Baer rings // J.
Pure and Appl. Algebra. – 2001. – 159. – P. 25 – 42.
4. Birkenmeier G. F., Kim J. Y., Park J. K. Principally quasi-Baer rings // Communs Algebra. – 2001.
– 29, # 2. – P. 639 – 660.
5. Mousssavi A., Javadi H. S., Hashemi E. Generalized quasi-Baer rings // Communs Algebra. –
2005. – 33. – P. 2115 – 2129.
6. Armendariz E. P. A note on extensions of Baer and p.p.-rings // J. Austral. Math. Soc. – 1974. –
18. – P. 470 – 473.
7. Kerempa J. Some examples of reduced rings // Algebra Colloq. – 1996. – 3, # 4. – P. 289 – 300.
8. Hong C. Y., Kim N. K., Kwak T. K. Ore extensions of Baer and p.p.-rings // J. Pure and Appl.
Algebra. – 2000. – 151. – P. 215 – 226.
Received 07.11.08
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
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| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| language | English |
| last_indexed | 2026-03-24T02:32:26Z |
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| publisher | Institute of Mathematics, NAS of Ukraine |
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| spelling | umjimathkievua-article-29002020-03-18T19:39:51Z Polynomial extensions of generalized quasi-Baer rings Поліноміальні розширення узагальнених квазіберових кілець Ghalandarzadeh, S. Javadi, H. S. Khoramdel, M. Галандарзадех, С. Яваді, Г. С. Хорамдель, М. In this paper, we consider the behavior of polynomial rings over generalized quasi-Baer rings and show that the generalized quasi-Baer condition on a ring R is preserved by many polynomial extensions. Розглянуто поведінку поліиоміальних кілець над узагальненими квазіберовими кільцями і показано, що узагальнена квазіберова умова щодо кільця R зберігається при багатьох поліпоміальїшх розширеннях. Institute of Mathematics, NAS of Ukraine 2010-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2900 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 5 (2010); 698–701 Український математичний журнал; Том 62 № 5 (2010); 698–701 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2900/2551 https://umj.imath.kiev.ua/index.php/umj/article/view/2900/2552 Copyright (c) 2010 Ghalandarzadeh S.; Javadi H. S.; Khoramdel M. |
| spellingShingle | Ghalandarzadeh, S. Javadi, H. S. Khoramdel, M. Галандарзадех, С. Яваді, Г. С. Хорамдель, М. Polynomial extensions of generalized quasi-Baer rings |
| title | Polynomial extensions of generalized quasi-Baer rings |
| title_alt | Поліноміальні розширення узагальнених квазіберових кілець |
| title_full | Polynomial extensions of generalized quasi-Baer rings |
| title_fullStr | Polynomial extensions of generalized quasi-Baer rings |
| title_full_unstemmed | Polynomial extensions of generalized quasi-Baer rings |
| title_short | Polynomial extensions of generalized quasi-Baer rings |
| title_sort | polynomial extensions of generalized quasi-baer rings |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2900 |
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