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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Інститут математики НАН України
2010
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irk-123456789-1661552020-02-19T01:25:50Z Polynomial extensions of generalized quasi-Baer rings Ghalanzardekh, S. Javadi, H.S. Khoramdel, M. Короткі повідомлення 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. 2010 Article Polynomial extensions of generalized quasi-Baer rings / S. Ghalanzardekh, H.S. Javadi, M. Khoramdel // Український математичний журнал. — 2010. — Т. 62, № 5. — С. 698–701. — Бібліогр.: 8 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166155 517.5 en Український математичний журнал Інститут математики НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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Короткі повідомлення Короткі повідомлення |
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Короткі повідомлення Короткі повідомлення Ghalanzardekh, S. Javadi, H.S. Khoramdel, M. Polynomial extensions of generalized quasi-Baer rings Український математичний журнал |
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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. |
| format |
Article |
| author |
Ghalanzardekh, S. Javadi, H.S. Khoramdel, M. |
| author_facet |
Ghalanzardekh, S. Javadi, H.S. Khoramdel, M. |
| author_sort |
Ghalanzardekh, S. |
| title |
Polynomial extensions of generalized quasi-Baer rings |
| title_short |
Polynomial extensions of generalized quasi-Baer rings |
| 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_sort |
polynomial extensions of generalized quasi-baer rings |
| publisher |
Інститут математики НАН України |
| publishDate |
2010 |
| topic_facet |
Короткі повідомлення |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/166155 |
| citation_txt |
Polynomial extensions of generalized quasi-Baer rings / S. Ghalanzardekh, H.S. Javadi, M. Khoramdel // Український математичний журнал. — 2010. — Т. 62, № 5. — С. 698–701. — Бібліогр.: 8 назв. — англ. |
| series |
Український математичний журнал |
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2025-07-14T20:49:06Z |
| last_indexed |
2025-07-14T20:49:06Z |
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1837656857462702080 |
| 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).
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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.
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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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