Vol. 19 (2018), No. 1, pp. 273–289 DOI: 10.18514/MMN.2018.2288
QUASIPOLARITY OF SPECIAL MORITA CONTEXT RINGS
O. G ¨URG ¨UN, S. HALICIOGLU, AND A. HARMANCI Received 03 April, 2017
Abstract. In this paper, we introduce a new concept of Morita context rings which we call special Morita context rings. We determine the conditions under which this kind of rings are quasipolar.
We use special Morita context rings to extend some results of quasipolar rings. Then many of the main results of quasipolar rings are special cases of our results for this general setting. Several basic characterizations and properties of these rings are given.
2010Mathematics Subject Classification: 16S50; 16S70; 16U99
Keywords: quasipolar ring, strongly clean ring, strongly-regular ring, Morita context ring, special Morita context ring
1. INTRODUCTION
Throughout this paper all rings are associative with identity and modules are unital.
The commutantandsecond commutantofa2Rare defined by comm.a/D fx 2 RjxaDaxg,comm2.a/D fx2RjxyDyxfor ally2comm.a/g, respectively.
U.R/,J.R/andN i l.R/will denote the set of all invertible elements, the Jacobson radical ofR and the set of all nilpotent elements, respectively. An element ain a ring R is called quasinilpotent if 1 ax2U.R/ for any x 2comm.a/. The set of all quasinilpotent elements ofRwill be denoted by QN.R/([9]). SetJ#.R/D fx2R j 9n2Nsuch thatxn2J.R/g. Obviously,J.R/J#.R/QN.R/and N i l.R/J#.R/QN.R/.
The notion of a quasipolar ring was introduced by Harte in his 1991 study on quas- inilpotent in rings. An elementa2Ris calledquasipolarprovided that there exists an idempotentp 2R such thatp 2comm2.a/, aCp2U.R/ andap2QN.R/.
A ring R is quasipolarin case every element in R is quasipolar. Any idempotent p satisfying the above conditions is called aspectral idempotent ofa. Koliha [11]
introduced the concept of a generalized Drazin inverse in a complex Banach algebra.
An elementaofRisgeneralized Drazin invertible[12] in case there is an element b2Rsatisfyingab2Db,b2comm2.a/anda2b a2QN.R/. Suchb, if it exists, is unique; it is called ageneralized Drazin inverseofaand will be denoted byagD. Koliha and Patricio [12] proved any quasipolar elementa2Rhas a unique spectral
c 2018 Miskolc University Press
idempotent denoted bya, andais quasipolar if and only ifais generalized Drazin invertible. Quasipolar rings have been studied by several authors [2,4,7,12,22].
Recall some definitions. A ring is calledstrongly cleanif every element is the sum of an idempotent and a unit which commute (see [16]). Following [1], an element a2Ris said to bestrongly-regularifan2anC1R\RanC1for somen2N. An elementa2Ris calledpseudopolarif there existsp2Dp2comm2.a/such thataC p2U.R/andap2J#.R/([21]). An elementaofRis(pseudo) Drazin invertible ([21]) [5] in case there is an elementb2Rsatisfyingab2Db,b2comm2.a/and (a2b a2J#.R/)a2b a2N i l.R/. Such b, if it exists, is unique; it is called a (pseudo) Drazin inverse ofa and will be denoted by (apD) aD. In 1958, Drazin showed that a is strongly -regular if and only if a has a Drazin inverse. Wang and Chen proved thatais pseudopolar if and only ifais pseudo Drazin invertible.
By definitions, we conclude that any strongly-regular element is pseudopolar, any pseudopolar element is quasipolar and any quasipolar element is strongly clean.
Morita contexts appeared as a key ingredient in the work of Morita that described equivalences between full categories of modules over rings with identities. Morita context rings form a very large class of rings generalizing matrix rings. One of the fundamental results in this direction says that the categories of left modules over the rings Aand B are equivalent if and only if there exists a strict Morita context connectingAandB. Other applications, though not stated in an explicit form, can be found in various places. A Morita context.A; B; M; N; ; /consists of two rings AandB, two bimodulesAMB,BNAand a pair of bimodule homomorphisms
WMN
BN!AandWNN
AM !B which satisfy the following associativity:
.m˝n/m0Dm.n˝m0/ and .n˝m/n0Dn .m˝n0/ (1.1) for any n; n02N, m; m02M. These conditions insure that the set of generalized matrices
a m n b
;a2A,b2B,m2M,n2N will form a ring, called the ring of theMorita context. A Morita context
A M
N B
withADBDR,M DN, and D D' is calleda special Morita context. ThroughoutS denotes the ring of a special Morita contextŒR; M; '.
The main purpose of this paper is to study quasipolarity of special Morita con- texts which are a natural generalization of (generalized) full matrix rings Ks.R/
orM2.RIs/
M2.R/. One of the motivations to study the concept of special Morita context rings is to construct nontrivial examples for quasipolar rings (see Examples1, 2, 3). In the present paper we use special Morita context rings to extend some res- ults of quasipolar rings (e.g. [2, Theorem 2.10], [10, Theorem 22], [19, Theorem 15, Theorem 18 and Theorem 22] and [21, Theorem 1.4]).
This paper is organized as follows. In Section 2, the forms of the Jacobson radical and the center of special Morita contexts are determined and some properties of this class of rings are investigated. Furthermore, we define (uniquely) weakly quasipolar rings and determine the relation between quasipolar rings and pseudopolar rings (or equivalently, uniquely strongly-rad clean rings). In particular, we show thatR is quasipolar if and only ifR is uniquely weakly quasipolar (Theorem2). This gives an affirmative answer to the question in [12, Remark 4.8] and [11, Lemma 2.4]. In Section 3, criteria are obtained for a single element ofS to be quasipolar for a local ringR. As a result, we see that (in Proposition5)S is quasipolar if and only ifS is weakly quasipolar whereM is uniquely bleached andR is local. In Section 4, we determine when a special Morita contexts over a commutative local ring is quasipolar.
It is shown thatS is quasipolar if and only ifx2 xCwD0is solvable for every w2I m'whereRis commutative local and Im'J.R/. This extend and improve many known results such as [10, Corollaries 11 and 12] and [19, Theorem 18]. In particular, we prove that ifR is commutative local and Im' is nilpotent, then S is quasipolar. This yields the main result of [19] (see Example1). Several equivalent conditions on quasipolar special Morita context rings over a (commutative) local ring are obtained.
In this paper, the ring of integers modulonis denoted byZn, and we writeMn.R/
andC.R/for the rings of allnnmatrices over the ringRand the set of all central elements ofR, respectively. For elementsa; bin a ringR, we use the notationa b to mean that a is similar to b, that is, b Du 1au for some u2U.R/. We set
M WJ.R/
D fm2M j'.mN
n/2J.R/for alln2Mgand J.R/WM
D fm2 M j'.nN
m/2J.R/for alln2Mg.
2. PRELIMINARY RESULTS
In this section, we decide the forms of the Jacobson radical and the center of spe- cial Morita contexts. We introduce (uniquely) weakly quasipolar rings and investigate relations between quasipolar rings and pseudopolar rings (or equivalently, uniquely strongly-rad clean rings). We prove thatRis quasipolar if and only ifRis uniquely weakly quasipolar (Theorem2). As a result we obtain an affirmative answer to the question in [12, Remark 4.8] and [11, Lemma 2.4].
The Jacobson radical formula of a Morita context was determined by Sands in his 1973 study on Radicals and Morita contexts (see [17]):
Theorem 1. LetRbe a ring. Then J.S/D
J.R/ M WJ.R/
M WJ.R/
J.R/
D
J.R/ J.R/WM J.R/WM
J.R/
:
Corollary 1. LetRbe a ring and Im'J.R/. Then J.S/D
J.R/ M M J.R/
:
Proposition 1. LetRbe a ring and Im'J.R/. IfJ.R/is nilpotent, thenJ.S/ is nilpotent.
Proof. Assume that J.R/t
D0for somet2N. By direct calculation one sees that
J.S/t
"
J.R/t
CI m' M
M
J.R/t
CI m'
#
I m' M M I m'
DWT. Further, we have
T2D
I m' I m'M I m'M I m'
, and soT2tD
"
I m't I m't
M I m't
M I m't
# D0:
Therefore
J.S/2t2
D0, that is,J.S/is nilpotent, as desired.
The following lemma is verified by direct calculation.
Lemma 1. LetRbe a ring. Then C.S/D
a 0 0 b
ja; b2C.R/andamDmb; bmDmafor allm2M
. Lemma 2. LetRbe a ring and Im'J.R/. Then
u m n v
is invertible inS if and only ifu; v2U.R/.
Proof. Since units always lift modulo the Jacobson radical, it is clear by Corol-
lary1.
Lemma 3. LetRbe a local ring and let˛2D˛2S. Then there existˇ; 2U.S/ such thatˇ˛ D
0 0
. Proof. Write˛ D
e m n f
wheree; f 2R andm; n2M. Since˛2D˛, we have
eDe2C'.m˝n/; mDe mCmf; nDneCf n; f Df2C'.n˝m/:
Ife2U.R/, then we see that 1 0
ne 1 1
e m n f
1 e 1m
0 1
D
e 0
0 f '.ne 1˝m/
:
Similarly, if f 2U.R/, then there exist ˇ; 2U.S/ such that ˇ˛ D
0 0
. Now assume thate; f 2J.R/. Then˛2J.S/, and so it is zero. We complete the
proof.
In view of [18, Theorem 4], we have the following result.
Corollary 2. Let R be a local ring and let ˛2D˛ 2S. Then there exists a ˇ2U.S/such thatˇ˛ˇ 1D
0 0
.
LetM be anR-R-bimodule and leta2R. la WM !M andra WM !M de- note, respectively, the abelian group endomorphisms ofM given byla.m/Damand ra.m/Dmafor allm2M.
Lemma 4. Let˛D
x 0 0 y
; XD
a m n b
2S. ThenX 2comm.˛/if and only ifa2comm.x/,b2comm.y/,m2ker.lx ry/andn2ker.ly rx/.
Proof. It is straightforward.
Corollary 3. Let˛D
1 0 0 0
2S andˇD
0 0 0 1
2S. Then comm.˛/D
a 0 0 b
ja; b2R
Dcomm.ˇ/:
Lemma 5. LetRbe a ring. Then
a m n b
7!
b n m a
is an automorphism of S.
Remark1. IfRis isomorphic to a ringS byf, thena2R is quasipolar if and only iff .a/is quasipolar inS.
Lemma 6. LetRbe a local ring. Then
a 0 0 b
2QN.S/if and only ifa; b2 J.R/if and only if
a 0 0 b
2J.S/.
Proof. IfRis a local ring, thenJ.R/DQN.R/, and so the proof is clear.
Wang and Chen prove that ([21, Theorem 1.4]) if there existsp2Dp2comm.a/
such thataCp 2U.R/ andakp2J.R/for some k2N, thenp is unique if and only ifp2comm2.a/. We extend this result as the following.
Theorem 2. LetRbe a ring and assume thatp2Dp2comm.a/,aCp2U.R/
andap2Rq ni l for somea; p2R. Thenp2comm2.a/if and only ifpis unique.
Proof. Ifp2comm2.a/, thenp is unique by [12, Proposition 2.3]. For the con- verse, supposep is unique and letxaDax for somex2R. Set c WDxp pxp, b D.aCp/ 1.1 p/. Then ab2 Db, 1 p Dab Dba, pCc DWe is an idem- potent,epDe andpeDp. We show thataCe2U.R/andae2QN.R/. Since .aCe/.bCp/D1CapCpx.1 p/.bCp/D1CpaCpxbD1Cp.aCxb/is
invertible if and only if1C.aCxb/pD1Capis invertible, andap2QN.R/,aCe is a right invertible element. Further, as.bCp/.aCe/D1CapCpx pxpD1C apCpxabD1Cap.1Cxb/is invertible if and only if1C.1Cxb/apD1Capis invertible,aCeis a left invertible element. HenceaCeis invertible. LetyaeDaey for somey2R. We prove that1Caey2U.R/. We know that1Caey 2U.R/if and only if1CayeD1Cayep2U.R/if and only if1CpayeD1Capye2U.R/
because epDe. Furthermore, pyeap Dpyepa DpyeaDpyae and appye D apyeDapeyeDpaeyeDpyaeeDpyae, we conclude that1Capye2U.R/as pa2QN.R/. ThuseDpCc Dp, and socD0becausep is unique. That is, we havexpDpxp. Analogously, it can be shown thatpxDpxp. ThereforexpDpx,
and sop2comm2.a/. The proof is completed.
Definition 1. LetR be a ring. Thena2R is calledweakly quasipolar provid- ing that there exists an idempotente2comm.a/ such thataCe2U.R/andae2 QN.R/. If this representation is unique, thena2Ris calleduniquely weakly qua- sipolar. A ring R is called (uniquely) weakly quasipolar if any element of R is (uniquely) weakly quasipolar.
Remark2. Lemma 2.4 in [11], Koliha proved that every weakly quasipolar element is quasipolar in a Banach algebra. But it is not true in a ring. LetRDZ2Œt1; t2; : : :.t1/
be a ring of polynomials in countably many indeterminates, localized at the prime ideal .t1/ (for details see [5, Example 2.4.3]). Let be the map which satisfies .ti/DtiC1. ThenRŒŒxI is the skew formal power series local ring overR. Con- siderAD
t2 0 0 t1
2M2 RŒŒxI
wheret12J RŒŒxI
, t22U RŒŒxI
. We directly see thatAis weakly quasipolar withE1D
0 x 0 1
,E2D
0 0 0 1
. ThereforeAis not quasipolar.
By Theorem2, we have the following result.
Corollary 4. LetRbe a ring. The following are equivalent fora2R.
(1) ais uniquely weakly quasipolar.
(2) ais quasipolar.
3. NONCOMMUTATIVE CASES
The goal of this section is to investigate quasipolarity of special Morita contexts over local rings. We begin with the following definition.
Definition 2. We say that a bimoduleRMRis called(bleached) cobleachedprovided that for anya2J.R/,b2U.R/, bothla rbandlb raare (surjective) injective on M. Further, a bimoduleRMRwill be calleduniquely bleachedif, for everyj2J.R/
andu2U.R/,la rbandlb raare surjective as well as injective.
Huang, Tang and Zhou [10, Corollary 6] proved that ifKs.R/is quasipolar, then Ris cobleached whereRis local. Our next result shows that the latter assumption is superfluous.
Theorem 3. LetRbe a ring. IfS is quasipolar, thenM is cobleached.
Proof. Letu2U.R/,j2J.R/, and let.lu rj/.m/D0wherem2M. Consider
˛D
u 0 0 j
2S. TakeED
0 0 0 1
. ThenE˛D˛E,˛CE is invertible and
˛E is quasinilpotent. Since˛ is quasipolar, by Theorem2, E must be unique and E2comm2.˛/. Note that
0 m 0 0
2comm.˛/. SoEcommutes with
0 m 0 0
,
which impliesmD0. As required.
Proposition 2. LetRbe a local ring. Then˛2S is strongly clean if and only if either˛2U.S/, orI2 ˛2U.S/, or˛is similar to
a 0 0 b
.
Proof. “)” Assume that˛ D
x y
´ t
is strongly clean. Then there exists an idempotent E 2S such that ˛ E 2U.S/ and E 2comm.˛/. By Corollary 2, there exists ˇ2U.S/ such thatˇ 1EˇD
e 0 0 f
. SinceR is local and E2D E, we have e2De; f2Df 2 f0; 1g. IfE D0, then ˛ 2U.S/. IfE DI2, then I2 ˛ 2U.S/. It follows that˛ is similar to
a 0 0 b
where a, b 2R because E2comm.˛/.
“(” If˛2U.S/orI2 ˛2U.S/, then it is strongly clean. Hence we can assume that˛is similar toˇD
a 0 0 b
. SinceRis local, there existe2De; f2Df 2R such thata e; b f 2U.R/,e2comm.a/andf 2comm.b/. Thereforeˇ F 2 U.S/andˇF DFˇ whereF D
e 0 0 f
. That is,ˇis strongly clean. It is well known that if x is strongly clean and xy, theny is strongly clean. Thus ˛ is
strongly clean, as asserted.
Proposition 3. LetRbe a local ring. Then the following statements are equivalent for˛D
x 0 0 y
2S. (1) ˛is quasipolar.
(2) Bothlx ryandrx lyare injective.
(3) comm.˛/D
a 0 0 b
ja; b2R
.
Proof. (1))(2) Suppose that.lx ry/.m/D0 for somem2M and˛CE 2 U.S/whereEDE2D
a b c d
2comm2.˛/and˛E 2QN.S/. By Lemma 4, we get
1 0 0 0
;
0 m 0 0
2comm.˛/. Then bD0Dc, a2Da andd2 Dd becauseE2DE2comm2.˛/. SinceRis local,a; d2 f0; 1g. In view of Corollary3, comm.E/is the set of all diagonal matrices inS. We conclude thatmD0; that is, lx ryis injective. Similarly, we see thatrx lyis injective. So holds (2).
(2))(1) Ifx; y2U.R/orx; y2J.R/, then we easily see that˛is quasipolar in S. Letx2J.R/andy2U.R/. WriteE D
1 0 0 0
. ThenE2DE2comm.˛/
and˛CE2U.S/. LetX D
a b c d
2comm.˛/. According to Lemma4,b 2 ker.lx ry/ and c 2ker.rx ly/. By (2), we have b D0Dc. Hence we get E2comm2.˛/by Corollary3. As˛ED
x 0 0 0
2J.S/by Theorem1, we have
˛E2QN.S/. That is,˛is quasipolar inS. (2))(3) LetX D
a m n b
2comm.˛/. By Lemma4, we havem2ker.lx
ry/andn2ker.rx ly/. By assumption, we getmDnD0, and socomm.˛/D a 0
0 b
ja; b2R
.
(3))(2) Suppose that.lx ry/.n/Dx n nyD0for somen2M. This gives thatX D
0 n 0 0
2comm.˛/. By (3), we havenD0and solx ry is injective.
Similarly, it can be proved thatrx lyis injective. Therefore we complete the proof.
Remark3. As is well known, ifab is quasipolar and.ab/gD Dc, then so isba and.ba/gDDbc2a[15] (or see [6]). In particular, ifais quasipolar andagDDc, thenu 1auDbis quasipolar for anyu2U.R/andbgDDu 1cu.
Proposition 4. Let R be a local ring. Then ˛ 2S is quasipolar if and only if
˛ 2U.S/, or˛2QN.S/, or˛ is similar to
a 0 0 b
where la rb, lb ra are injective.
Proof. “)” Assume that˛ D
x y
´ t
is quasipolar with spectral idempotent E D
p m n q
. Then, by Corollary 2, there existsˇ2U.S/such thatˇ 1EˇD e 0
0 f
. SinceRis local andE2DE, we havee2De; f2Df 2 f0; 1g. IfED0,
then˛2U.S/. IfEDI2, then˛2QN.S/. It follows that˛is similar to
a 0 0 b
wherea,b2RbecauseE2comm2.˛/. In view of Proposition3,la rb,lb ra
are injective, as asserted.
“(” It is clear from Proposition3and Remark3.
Remark 4. Clearly, if a2U.R/[QN.R/, then it is weakly quasipolar. In this case, we say thataistrivial weakly quasipolar. Ifais not inU.R/\QN.R/, we say thataisnon-trivial weakly quasipolar.
Theorem 4. LetRbe a local ring. Assume that˛is non-trivial weakly quasipolar withE2DE2S. The following are equivalent.
(1) Eis unique.
(2) E2comm2.˛/.
(3) ˛is similar to
a 0 0 b
wherela rb,lb ra are injective.
Proof. By Theorem2, we conclude that (1),(2).
(2))(3) By assumption, we get˛ is quasipolar. Then˛is similar to
a 0 0 b
wherela rb,lb raare injective by Proposition4.
(3))(1) In view of Proposition4,˛is quasipolar, and soE is unique.
Theorem 5. LetRbe a local ring. For anyu2U.R/andj 2J.R/, the following are equivalent.
(1) ru lj is injective andlu rj is an isomorphism.
(2) For eachm2M,˛ D
u m 0 j
is quasipolar in S with˛ D
0 n 0 1
for somen2M.
Proof. (1))(2) Letm2M. Sincelu rj is an isomorphism, there existsn2M such thatun nj D m. WriteED
0 n 0 1
. This givesE2comm.˛/. As
.˛CE/
u 1 u 1.mCn/.jC1/ 1 0 .jC1/ 1
D
u 1 u 1.mCn/.jC1/ 1 0 .jC1/ 1
;
.˛CE/DI2, we have ˛CE 2U.S/. We show thatE 2comm2.˛/. Let X D x y
´ t
2comm.˛/. This implies that´D0,tj Djt,uxDxuanduy yj D
xm mt becauseru lj is injective. We conclude that
u.yCx n nt / .yCx n nt /j Duy yjCux n unt x njCntj Duy yjCxun unt x njCnjt Dxm mtCxun unt x njCnjt Dxm mtCx.un nj /C.nj un/t Dxm mtCx. m/Cmt
D0:
Therefore XE DEX because lu rj is injective. Finally, we show that ˛E 2 QN.S/. Assume AD
a b c d
2comm.˛E/. ThenI2C˛EA2U.S/because
˛ED
0 nj 0 j
, as desired.
(2))(1) IfmD0, thenlu rj,ru lj are injective. It remains to prove thatlu rj
is surjective. That is, for eachm2M, there existsn2M such thatun nj Dm. Let m2M. We write˛D
u m 0 j
. By (2), there exists an idempotentED
0 n 0 1
such thatE2comm2.˛/,˛CE 2U.S/, and˛E2QN.S/. Since˛E DE˛, we
getu. n/ . n/j Dm.
By a similar method of the proof of Theorem5, we can derive the following.
Theorem 6. LetRbe a local ring. For anyu2U.R/andj 2J.R/, the following are equivalent.
(1) lu rj is injective andru lj is an isomorphism.
(2) For anym2M,˛D
u 0 m j
is quasipolar inS with˛D
0 0 n 1
for somen2M.
According to Theorems5and6, the following result is immediate.
Corollary 5. LetRbe a local ring. The following statements are equivalent.
(1) M is uniquely bleached.
(2)
R M 0 R
or
R 0
M R
is quasipolar.
Proposition 5. LetR be a local ring andM is cobleached. Then the following are equivalent.
(1) ˛2S is quasipolar.
(2) ˛2S is weakly quasipolar.
Proof. (1))(2) It is clear.
(2))(1) Assume thatE2DE2comm.˛/,˛CE2U.R/and˛E2QN.S/.
IfEis zero or identity inS, then˛is quasipolar. Hence, we deduce thatE is similar to the diagonal matrix by Corollary2. Thus, we have ˛ is similar to the diagonal matrix. SinceM is cobleached, we get that˛is quasipolar by Theorem4.
4. COMMUTATIVE CASES
LetRbe a commutative ring and˛D
a m n b
2S. In this section, we assume that'satisfying symmetry; that is,'.m˝n/D'.n˝m/for allm; n2M. We define det .˛/Dab '.m˝n/andt r.˛/DaCb, andr˛D
ra rm rn rb
forr2R. Note that ifRis commutative andM is rightR-module, thenM is leftR-module where rmWDmrfor allr2R,m2M. Therefore,M is uniquely bleached.
Lemma 7. LetRbe a commutative ring and let˛; ˇ2S. The following hold:
(1) det .˛ˇ/Ddet .˛/det .ˇ/.
(2) ˛2U.S/if and only ifdet .˛/2U.R/.
(3) If˛ˇ, thendet .˛/Ddet .ˇ/andt r.˛/Dt r.ˇ/.
(4) ˛2 t r.˛/˛Cdet .˛/I2D0.
(5) det .I2C˛/D1Ct r.˛/Cdet .˛/.
Proof. It is straightforward.
Theorem 7. LetR be a commutative ring and let˛2S. Thendet .˛/,t r.˛/2 J.R/if and only if˛is quasinilpotent inS.
Proof. Assume thatdet .˛/,t r.˛/2J.R/and let˛D
a m n b
2S. By Lemma7(4), we have˛2Dt r.˛/˛ det .˛/I22J.S/, and so˛2QN.S/. Con- versely, suppose that ˛ 2 QN.S /. To prove that det .˛/2 J.R/, let y 2R. It is easy to check that ˇ˛ D˛ˇDdet .˛/I2 where ˇD
b m n a
2S. Since
˛2QN.S/, we have thatI2Cyˇ˛2U.S/, and so1Cydet .˛/2U.R/. There- fore, we get det .˛/2J.R/. Let x 2R. Then we have I2Cx˛2U.S/, and so det .I2Cx˛/D1Cxt r.˛/Cx2det .˛/2U.R/ by Lemma 7(2). This gives that 1Cxt r.˛/2U.R/; thus,t r.˛/2J.R/. The proof is completed.
Corollary 6. LetR be a commutative ring. Then the following statements are equivalent.
(1) ˛2QN.S/.
(2) det .˛/,t r.˛/2J.R/.
(3) ˛22J.S/.
(4) ˛k2J.S/for somek2.
Remark5. LetRbe a commutative ring. Then we conclude that QN.S/DJ#.S/D f˛2S j˛22J.S/g:
Recall that an elementain a ringRis calledstrongly-rad cleanif there exists an idempotente2comm.a/such thata e2U.R/and.ae/k2J.R/for some integer k.
Corollary 7. LetRbe a commutative local ring. The following are equivalent.
(1) ˛is pseudopolar inS.
(2) ˛is quasipolar inS.
(3) ˛is weakly quasipolar inS. (4) ˛is strongly-rad clean inS.
Proof. (1))(2), (2)) (3) and (4))(3) are clear. By Remark5, we deduce that (3))(4) and (2))(1). According to Proposition5, we have (3))(2).
Theorem 8. LetRbe a commutative local ring and let˛2S. Then the following statements are equivalent.
(1) ˛is quasipolar inS.
(2) ˛2U.S/, or˛2QN.S/, orx2 t r.˛/xCdet .˛/D0is solvable.
Proof. (1)) (2) Let ˛2S be quasipolar. We may assume that ˛ …U.S/and
˛ …QN.S/. Then ˛ is similar to ˇD
a 0 0 b
where a2U.R/, b2J.R/ by Proposition4. According to Lemma7(3),t r.˛/Dt r.ˇ/anddet .˛/Ddet .ˇ/. This givesx2 t r.˛/xCdet .˛/Dx2 t r.ˇ/xCdet .ˇ/. Sincea2 t r.ˇ/aCdet .ˇ/D 0, the equationx2 t r.˛/xCdet .˛/D0is solvable inR.
(2) ) (1) If ˛ is invertible or quasinilpotent in S, then ˛ is quasipolar. Let
˛D
a11 a12
a21 a22
2S and supposex2 t r.˛/xCdet .˛/D0has rootsa; b2R.
Since ˛…U.S/\QN.S/, we get det .˛/Da11a22 '.a12˝a21/Dab2J.R/
andt r.˛/Da11Ca22DaCb2U.R/by Theorem7. So one of a; bmust be in U.R/and the other must be inJ.R/. By Lemma5 and Remark1, we can assume thata2U.R/, b2J.R/ and a11 2U.R/. Let ˇD
1 0 a21.a a22/ 1 1
. By Lemma7(2),ˇ2U.S/and easy calculation shows thatˇ 1˛ˇis an formal triangu- lar matrix in
R M 0 R
. Therefore˛is quasipolar from Corollary5and Remark3.
We complete the proof.
Now we extend [19, Theorem 15] as follows.
Theorem 9. LetRbe a commutative local ring and let˛2S. Then the following statements are equivalent.
(1) ˛is non-trivial strongly clean inS.
(2) det .˛/2J.R/,t r.˛/21CJ.R/andx2 t r.˛/xCdet .˛/D0is solvable.
Proof. (1))(2) It is clear by Proposition2and Lemma7.
(2))(1) Sincedet .˛/2J.R/andt r.˛/21CJ.R/, we deduce that˛…QN.S/\ U.S/. By Theorem8,˛is quasipolar, and so˛ is strongly clean.
Remark6. According to Theorem9,˛2Sis non-trivial strongly clean if and only if it is non-trivial quasipolar inS whereRis commutative local.
Theorem 10. LetRbe a commutative local ring. Then the following are equival- ent.
(1) S is quasipolar.
(2) For every˛2S withdet .˛/2J.R/, one of the following holds:
(i) t r.˛/2J.R/,
(ii) x2 t r.˛/xCdet .˛/D0is solvable inR.
Proof. (1))(2) Suppose that˛2S withdet .˛/2J.R/. By.1/, there exists an idempotent˛2S such thatE2comm2.˛/and˛CE2U.S/andE˛2QN.S/.
IfEDI2, then˛2QN.S/and sot r.˛/2J.R/by Theorem7. So we can assume that˛ …QN.S/. Since det .˛/2J.R/, ˛…U.S/ by Lemma7(2). According to Theorem8, the equationx2 t r.˛/xCdet .˛/D0is solvable inR.
(2))(1) Let ˛2S. If det .˛/2U.R/, then˛ 2U.S/ and so˛ is quasipolar.
Letdet .˛/2J.R/. Ift r.˛/2J.R/, then˛is quasipolar by Theorem7. Hence we assume thatt r.˛/2U.R/. This gives˛…U.S/and˛…QN.S/. By Theorem8,˛
is quasipolar and soS is quasipolar.
Lemma 8. LetRbe a ring and letu2U.R/\C.R/. Thena2Ris quasipolar ring if and only ifuais quasipolar inR.
Proof. “)” It follows from [3, Lemma 2].
“(” Assume thatauDsCq wheresis strongly regular,s2comm2.au/,q 2 QN.R/ andsqDqsD0 by [7, Corollary 2.17]. Then we getaDu 1sCu 1q.
It can be shown thatu 1s2comm2.a/ andu 1su 1qDu 1qu 1sD0because u2C.R/. Sincesis strongly regular, there existst2comm2.s/such thatsDs2t. Multiplying by u 1 yields u 1sDu 1s2t D
u 1s2
ut. This gives that u 1s is strongly regular inR. To proveu 1q2QN.R/, letxu 1qDu 1qx. Asuis central, we deduce thatu 1xqDqu 1xDu 1qx, and so1Cu 1qx2U.R/becauseq2
QN.R/, as asserted.
Theorem 11. LetRbe a commutative local ring. The following are equivalent.
(1) S is pseudopolar.
(2) S is quasipolar.
(3) S is weakly quasipolar.
(4) S is strongly-rad clean.
(5) S is strongly clean.
Proof. It suffices to show that (5))(2). Let˛D
a m n b
2S. We can further assume that˛…U.S/\QN.S/. Then we havedet .˛/2J.R/andt r.˛/2U.R/.
Case I.Let'.m˝n/2J.R/. Then we may assumea2U.R/andb2J.R/by Lemma5. We writea 1˛D
1 p q j
wherea 1nDq,a 1mDpanda 1bDj. Note thata 1I2is central invertible inS by Lemma1. Sincej 2J.R/andjC '.p˝q/
DWw2jCI m', we havedet .a 1˛/2J.R/anddet .I2 a 1˛/2J.R/.
This impliesa 1˛is non-trivial strongly clean. Thusx2 t r.a 1˛/Cdet .a 1˛/D 0 is solvable by Theorem 9, and so a 1˛ is quasipolar, and so˛ is quasipolar by Lemma8.
Case II.Let'.m˝n/DWu2U.R/. Now we prove thatt2 t wD0is solvable for allw2J.R/. Letw2J.R/and writeˇD
0 u 1m
nw 1
. Sincedet .ˇ/D0 '.u 1m˝nw/D u 1w'.m˝n/D u 1wuD w2J.R/anddet .I2 ˇ/D 1 t r.ˇ/Cdet .ˇ/D w2J.R/,ˇis a non-trivial strongly clean element, and so t2 t wD0is solvable by Theorem9. Hencet2 tCdet .˛/t r.˛/2 D0is solvable. This gives thatx2 t r.˛/xCdet .˛/D0is solvable. Consequently,˛ is quasipolar. We
complete the proof.
Remark 7. According to the proof Theorem11, we deduce that ifS is strongly clean and Im'\U.R/¤¿, thenx2 x wD0is solvable for allw2J.R/.
Theorem 12. LetRbe a commutative local ring with Im'J.R/and let˛2S. Then the following are equivalent.
(1) S is quasipolar.
(2) x2 .1Cj /xCwD0is solvable for anyj 2J.R/andw2jCI m'.
(3) x2 xCwD0is solvable for anyw2I m'.
Proof. (1))(2) Let˛D
1 m n j
wherej 2J.R/,wDjC'.m˝n/. By (1),˛is quasipolar. Sincedet .˛/DjC'.m˝n/2J.R/andt r.˛/D1Cj2U.R/, we have thatx2 .1Cj /xCwD0is solvable by Theorem10(2).
(2))(1) Let˛D
a m n b
2S. Ifa; b2J.R/ora; b2U.R/, then˛2J.S/ or ˛ 2U.S/, and so ˛ is quasipolar. Hence, by Lemma 5, we may assume that a2U.R/andb2J.R/. Then we conclude thata 1˛D
1 p q j
wherea 1nDq, a 1mDpanda 1bDj. Sincej 2J.R/andjC '.p˝q/
DWw2jCI m', by assumption, we getx2 .1Cj /xCwD0 is solvable. This gives thata 1˛ is quasipolar, and so˛is quasipolar by Lemma8.
(1))(3) Similar to the proof of (1))(2).
(3) ) (1) Let˛D
a m1
n1 b
2S. If a; b2J.R/ ora; b2U.R/, then ˛ 2 J.S/ or˛ 2U.S/, and so ˛ is quasipolar. Hence, by Lemma 5, we may assume that a2U.R/ and b2J.R/. Then we conclude that a 1˛ D
1 m2
n2 j
DWˇ wherea 1n1Dn2,a 1m1Dm2anda 1bDj. It is well known thatˇis strongly clean if and only if I2 ˇ is strongly clean. Hence, we now consider I2 ˇD 0 m2
n2 1 j
. We directly see that.1 j / 1.I2 ˇ/D
0 m n 1
wheremD .1 j / 1m2 andnD .1 j / 1n2. By (3), we have thatx2 x '.m˝n/D0 is solvable, and so.1 j / 1.I2 ˇ/is (non-trivial) quasipolar. In view of Lemma8, I2 ˇ is (non-trivial) quasipolar; henceˇis (non-trivial) quasipolar by Remark 6.
Therefore, by Lemma8, we deduce that˛is quasipolar.
The following theorem is a generalization of Theorem 22 in [19].
Theorem 13. LetRbe a commutative local ring. If Im'is a nilpotent ideal ofR, thenS is quasipolar.
Proof. We can assume that (Im'/tD0for somet2N. In this case, we also note that Im'J.R/. By Theorem12, it suffices to prove thatx2 xCwD0is solvable for anyw2I m'. LetwD'.m˝n/andf0.x/Dx2 xCwD0. We set
WkDn
wkx2 uxCwD02RŒxju21CJ.R/; w2J.R/o
for anyk0. Then we havef0.wCwx/Dwf1.x/wheref1.x/Dwx2CxCw2 W1. Further, we see that f1. wCwx/Dwf2.x/where f2.x/Dw2x2 .2w2 1/xCw22W2. By iteration of this process, we getfk 1.wkCwx/Dwfk.x/for allkD1; : : : ; t. Therefore,
ft.at/2R;
) ft 1.at 1/2wR;
) ft 2.at 2/2w2R;
:::
) f1.a1/2wt 1R;
) f0.a0/2wtRD0:
So we see thatx2 xCwD0is solvable. We complete the proof.
We wind up the paper with some examples of special Morita contexts and quasi- polar rings. In spite of the fact that it is difficult to find a pair of bimodule homo- morphisms which satisfy the associativity rule, we supply the following examples.
Example1. Let DŒM; N; A; B; ; be a Morita context rings. TakeM DND RDADB and D DW' is defined bya˝b7!sab(ora˝b7!s2ab) where
sis central inR. ThenS is a generalized matrix ring overRdenoted byKs.R/(or M2.RIs2/) (for more details see [8,10,13,14,19,20]). It is easy to see thatS is isomorphic toM2.R/whensD1. Therefore, we see that the special Morita context rings are a natural generalization of generalized matrix ring.
Example2. LetRbe a local ring with Im'D0. ThenS is quasipolar if and only ifS is strongly clean andM is cobleached (if and only ifM is uniquely bleached).
Proof. “)” It is clear.
“(” Assume thatS is strongly clean andM is cobleached. Firstly, we prove that M is bleached. Letm2M,u2U.R/andj2J.R/. ConsiderˇD
u m 0 j
2S. By assumption,ˇis strongly clean. That is, there exists an idempotentE such that ˇ E is invertible andE 2comm.ˇ/. Take E D
x y
´ t
. Since E2DE and ˇEDEˇ, we directly see thatx2Dx,t2Dt,xuDux,tj Djt anduyCmtD xmCyj, and hencexD0,tD1anduy yj D mbecauseRis local andˇ E2 U.S/. ThusM is uniquely bleached. Let˛D
a m n b
. It is enough to show that
˛is weakly quasipolar by Proposition5. Without loss of generality, we may assume that a2U.R/ and b 2J.R/. AsM is bleached, there exist x; y 2M such that ux xj D mandjy yuDn. WriteE D
0 x y 1
. This gives thatE2DE, E2comm.˛/and˛E2J.S/, and so˛is weakly quasipolar.
Example3. We consider the special Morita contextSD
R M
M R
whereRD Z4,MDZ2˚Z2and'WM˝M!Z4is defined by' .a; b/˝.c; d /
D2.acCbd /.
This implies that'satisfy the associativity conditions and Im'D f0; 4g, and so Im' is nilpotent ideal inR. Hence, by Theorem13,SDŒR; M; 'is quasipolar.
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Authors’ addresses
O. G ¨urg ¨un
Ankara University, Department of Mathematics, 06100 Ankara, Turkey E-mail address:orhangurgun@gmail.com
S. Halicioglu
Ankara University, Department of Mathematics, 06100 Ankara, Turkey E-mail address:halici@ankara.edu.tr
A. Harmanci
Hacettepe University, Department of Mathematics, 06800 Ankara, Turkey E-mail address:harmanci@hacettepe.edu.tr
