SHEN Hongdi, CHEN Huanyin

(School of Science, Hangzhou Normal University， Hangzhou 310036, China)

On UnitJ-clean Rings

SHEN Hongdi, CHEN Huanyin

(School of Science, Hangzhou Normal University， Hangzhou 310036, China)

An elementa∈Rright (left) unitJ-clean if there is a unitu∈Rsuch thatau(ua) isJ-clean. A ringRis called right (left) unitJ-clean if each element is right (left) unitJ-clean. In this paper, we get the results that everyJ-clean ring is unitJ-clean, every unitJ-clean is unit clean and every 2-good ring is unit clean but the converse of all the three conclusions are not true. Further, we prove that for a unitJ-clean ringRis 2-good if and only if 1=u+vfor someu,v∈U(R). Also whenRis an abelian ring,Iis an ideal ofRandI?J(R), thenRis unitJ-clean if and only if (1)R/Iis unitJ-clean. (2)Idempotents lift moduloJ(R).

J-clean ring; unitJ-clean ring; idempotent; Jacobson radical

1 Introduction

In [1] the author introduce unit clean rings. A ringRis clean if every elementa∈Rcan be written in the form ofa=e+uwhereeis an idempotent anduis a unit. This concept was extended to unit clean ring in [1]. An elementa∈Rright (left) unit clean if there is a unitu∈Rsuch thatau(ua) is clean. A ringRis called right (left) unit clean if each element is right (left) unit clean. Many properties of such rings are studied in[1]. Inspired by this article and combining the notion ofJ-clean (A ring is calledJ-clean if each elementa∈Rcan be written in the form ofa=e+jwhereeis an idempotent andjis a Jacobson radical. ) We call an elementa∈Rright (left) unitJ-clean if there is a unitu∈Rsuch thatau∈R(ua∈R) isJ-clean. A ringRis called right (left) unitJ-clean if each element is right (left) unitJ-clean.

In this article we also use some related notion such as n-good ring and so on. We call a ringRis a n-good ring if each elementa∈Rcan be presented asa=u1+u2+u3+……+unwhereui∈U(R) for each 1≤i≤n,i∈Z. An elementr∈Ris called unit regular if there exists a unituinRsuch thatrur=r.

In this paper, we get the results that everyJ-clean ring is unitJ-clean, every unitJ-clean is unit clean and every 2-good ring is unit clean but the converse of all the three conclusions are not true. Further, we prove that for a unitJ-clean ringRis 2-good if and only if 1=u+vfor someu,v∈U(R). Also whenRis an abelian ring,Iis an ideal ofRandI?J(R), thenRis unitJ-clean if and only if (1)R/Iis unitJ-clean (2)Idempotents lift moduloJ(R).

Throughout this paper, all rings are associative rings with an identity.Id(R) denotes the idempotents ofR,J(R) denotes the Jacobson radical ofR,U(R) denotes the unit ofR, Ureg(R) represents the unit regular elements and we useTn(R) to stand for the ring of alln×nupper triangular matrices over a ringR.

2 Equivalent Characterizations

Definition 1 A ringRis called a right (left) unitJ-clean ring if for every elementa∈Rthere is a unitu∈Rsuch thatau=e+j(ua=e+j) wheree∈Id(R) andj∈J(R).

Lemma 1 An elementa∈Ris right unitJ-clean if and only if it is left unitJ-clean, and then we call it unitJ-clean.

Proof Letabe a right unitJ-clean element then there exists a unitusuch thatau=e+jwheree∈Id(R) andj∈J(R). Thusa=eu-1+ju-1. We letc=eu-1, thencu=e,cucu=cu,cuc=c, we multiplyuby the left then we get the result thatucuc=uc, thusucis also an idempotent, we note it byf, soc=u-1f,a=c+ju-1=u-1f+ju-1,ua=f+uju-1wheref∈Id(R) anduju-1∈J(R) we can see that it is left unitJ-clean.

The converse can be proved in a similar way.

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In [1] The following simple fact is known, we include a proof for readers’ convenience.

Lemma 2 An elementa∈Ris unit regular if and only if it can be written in the form ofa=euwheree∈Id(R) andu∈U(R).

Proof Letabe unit regular thenaua=awhereuis a unit. We imply thatau=eis an idempotent by the formula given in the front. Thusa=eu-1wheree∈Id(R) andu-1∈U(R). If we assume thata=eu, thenau-1=eand thusau-1au-1=au-1. Multiplying on the right byugives us thatau-1a=aandais unit regular.

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Theorem 1 A ringRis unitJ-clean if and only if every elementainRcan be written in the form ofa=r+jwherer∈Ureg (R) andj∈J(R).

Proof LetRbe a unitJ-clean ring then for every elementa∈Rthere exists a unitusuch thatau=e+jwheree∈Id(R) andj∈J(R). Thena=eu-1+ju-1whereeu-1∈Ureg(R) by Lemma 2 andju-1∈J(R). Conversely, if for every elementa∈R,a=r+jwherer∈Ureg(R) andj∈J(R), thena=eu+jsinceris an unit regular element and it can be replaced byeufor some idempotenteand some unituby Lemma 2. Soau-1=e+ju-1whereeis an idempotent andju-1belongs to Jacobson radical. Thus we get the result thatRis unitJ-clean.

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Example 1 EveryJ-clean ring is unitJ-clean, but the converse is not true.

Proof It is obvious that everyJ-clean ring is unitJ-clean as we can just takeu=1. However the converse is not true. We take a division ring which is not isomorphic to Z2. As we all know that in a division ring idempotents are only 0 and 1 and there is only one element 0 in the Jacobson radical. So there must be some elements can not be presented as the form ofa=e+jwheree∈Id(R) andj∈J(R) if there are more than two elements in a division ring. More clearly, Z3is a division ring and 2∈Z3can not be written in the form ofe+jsince 0+0=0 and 1+0=1. Cheerly, we find that every division ring is unitJ-clean. Ifa∈Randa=0,au=0+0 whereuis a unit and 0 is both idempotent and Jacobson radical. On the other hand ifa≠0, thenamust be a unit, thus we can writeain the form thataa-1=1+0 wherea-1is a unit and 1∈Id(R), 0∈J(R). So we get the result.

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Theorem 2 Every unitJ-clean ring is unit clean.

Proof For any elementa∈R,a-1=r+jwherer∈Ureg(R) andj∈J(R) sinceRis unitJ-clean. Thena=r+1+jwherer∈Ureg(R) and (1+j)∈U(R), thus it is unit clean by [1,Lemma5.2].

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However, the converse is not true. [1,Proposition5.3] gave out an example that is unit clean but not clean, now we will rewrite the example here and give the proof that it is unit clean but also not unitJ-clean.

Example 2 LetRbe an indecomposable commutative ring. IfRhas exactly two maximal ideals and 2 is a unit inR, thenRis unit clean but not unitJ-clean.

Proof Firstly, we will show it is unit clean. We noteM1andM2to represent two maximal ideals separately. For any elementa∈Rwe have four cases. (1)ais a unit, (2)a∈M1∩M2=J(R), (3)a∈M1M2and (4)a∈M2M1.

In the first case, we writea=0+asinceais a unit.

In the second case, we know that 1-av=ufor someuandvinU(R) sinceais in Jacobson radical, that isav=1-uwhere 1∈Id(R) and -u∈U(R), soais unit clean.

In the third case,a∈M1M2, thena+1 anda-1 are not belonging toM1. Since otherwise, 1 and -1 would be elements ofM1. If botha+1 anda-1 ∈M2, then (a+1)+(a-1)=2a∈M2. Thusa∈M2since 2 is a unit, a contradiction. So eithera+1 ora-1 is a unit, thusa=-1+u,a·(-1)=1-uora=1+u, it is obvious thatais unit clean.

Since the third and fourth cases are symmetric, we can prove the fourth case in the same way.

Next we will show that it is not unitJ-clean. We assume it is unitJ-clean, then for any elementainRwe can write it in the form ofa=eu+j. SinceRis an indecomposable commutative ring, the only idempotents we have are 0 and 1, thusa=0+jora=u+jthat isais either a Jacobson radical or a unit, we get thatRis local, a contradiction. SoRis not a unitJ-clean ring.

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Theorem 3 Every 2-good ringRis a unit clean ring.

Proof For any elementa∈R,a=u+vwhereu,v∈U(R), thenau-1=1+vu-1where 1 is an idempotent andvu-1is a unit, so it is unit clean.

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From the preceding result, we may ask that if every 2-good ring is unitJ-clean? If we can find a 2-good ring that is not unitJ-clean then we give out another example that a ring is unit clean but not unitJ-clean.

Example 3 In [2,Propsition 6] the author give the result that a proper matrix ring over an elementary divisor ring is 2-good. Every Euclidean domain proper matrix rings are strongly 2-good. We takeR=M2(Z) for instance. As we all know that Z is an Euclidean domain, soR=M2(Z) is 2-good. HoweverR/J(R)?R(J(R)=0) is not regular of course not unit regular whileR/J(R) is unit regular is a necessary condition for a unitJ-clean ring. Thus we solve the problem.

Lemma 3 Every unit regular ring in which 1 is the sum of two units is a 2-good ring.

Proof We get the result in [5].

Theorem 4 LetRbe a unitJ-clean ring. Then the following are equivalent:

(1)Ris 2-good;

(2)1=u+vfor someu,v∈U(R).

Proof (1)?(2) It is obvious.

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Corollary 1 LetRbe a unitJ-clean ring and 2∈U(R), then R is 2-good.

Also,wecanprovethiscorollarybyTheorem4, 1=1/2+1/2where1/2isaunit,sowegetit.

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Theorem5Inalocalringeveryunitregularelementiseitheraunitor0.

ProofForanyelementa∈Ureg(R),ifa∈U(R),thenitisaunit,ifa∈J(R),thena=eu, au-1=e∈J(R).Thuswegetthate=0sinceidempotentinJacobsonradicalmustbe0.Soa=0u=0.

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Proposition1LetRbeaunitJ-cleanringthenanyelementa∈J(R)canbepresenteduniquelyasa=0+awhere0∈Ureg(R)anda∈J(R).

ProofForanyelementa∈J(R), a=eu+jsinceRisaunitJ-cleanring.Thenau-1=e+ju-1, e=au-1-ju-1∈J(R)sincebothau-1andju-1belongtoJacobsonradical,soe=0anda=0u+j=0+jistheuniquepresentationofa.

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Proposition2EveryhomomorphicimageofaunitJ-cleanringisunitJ-clean.

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Theorem6LetRbeanabelianring. IisanidealofRandI?J(R).ThenRisunitJ-cleanifandonlyif(1)R/IisunitJ-clean; (2)IdempotentsliftmoduloJ(R).

ProofR/IisthehomomorphicimagineofR,soitisunitJ-cleansinceRisunitJ-cleanbyProposition2.

Corollary2LetRbeanabelianring.ThenRisunitJ-cleanifandonlyif

(1)R/J(R)isunitJ-clean; (2)IdempotentsliftmoduloJ(R).

ProofItisobviousbyTheorem6.

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Corollary3LetRbeanabelianring.ThenRisunitJ-cleanifandonlyif

(1)R/J(R)isunitregular; (2)IdempotentsliftmoduloJ(R).

ProofOnedirectionisobvious. R/J(R)isunitregularsinceRisunitJ-clean.WehaveproveIdempotentsliftmoduloJ(R)byTheorem6.

Conversely,if(1)and(2)hold.AsweknowunitregularringisunitJ-cleanwecangettheresultbyCorollary2.

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Itisobviousthatabovethreeconclusionsarerightforcommunicativering.

Theorem7LetRbeaunitJ-cleanringwithtwomaximalideals,thenRmustcontainnontrivialidempotent.

ProofForanyelementa∈R, a=eu+jforsomee∈Id(R), u∈U(R)andj∈J(R)sinceRisaunitJ-cleanring.Ifa∈J(R), a=eu+j, eu=a-j, e=(a-j)u-1∈J(R),sowegete=0.Ifa∈U(R) a=eu+j, eu=a-j, e=(a-j)u-1∈U(R),thene=1.Ifa∈M1M2, a=eu+j,nowweassumethateisatrivialidempotent,ife=0,thena=j∈J(R),acontradiction.ife=1,thena=u+j∈U(R),alsoacontradiction.Wegettheresultthatemustbeanontrivialidempotent.Wecandiscussa∈M2M1inasimilarway.

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LetP(R)betheprimeradicalofR,i.e.,theintersectionofallprimeideals.ItisobviousthatP(R)?J(R)sinceeverymaximalidealisprimeideal.

Definition2AringRiscalledaunitP-cleanifforeveryelementa∈Rthereisaunitu∈Rsuchthatau=e+p (ua=e+p)wheree∈Id(R)andp∈P(R).

Theorem8AringRisunitP-cleanifandonlyifeveryelementainRcanbewrittenintheformofa=r+pwherer∈Ureg(R)andp∈P(R).

ProofLetRbeaunitP-cleanringthenforeveryelementa∈Rthereexistsaunitusuchthatau=e+pwheree∈Id(R)andp∈P(R).Thena=eu-1+pu-1whereeu-1∈Ureg(R)byLemma2andpu-1∈P(R).Conversely,ifforeveryelementa∈R, a=r+pwherer∈Ureg(R)andp∈P(R),thena=eu+psincerisaunitregularelementanditcanbereplacedbyeuforsomeidempotenteandsomeunitubyLemma2.Soau-1=e+pu-1whereeisanidempotentandpu-1belongstoP(R).ThuswegettheresultthatRisunitP-clean.

Theorem9LetRbearing.ThenRisunitP-cleanifandonlyif

(1)P(R)=J(R); (2)RisunitJ-clean.

ProofAsweallknowthatP(R)?J(R),whatweshoulddoistoprovethatJ(R)?P(R).Foranyelementa∈J(R),wehavea=eu+psinceRisunitP-clean.Thuseu=a-p∈J(R)sincea∈J(R)andp∈P(R)?J(R),thene∈J(R), e=0.Wegeta=p∈P(R), J(R)?P(R), J(R)=P(R).AsP(R)?J(R),itisobviousthatunitP-cleanisunitJ-clean.

Conversely,assumethat(1)and(2)hold.Theconclusionisobvious.

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Theorem10AringRisunitP-cleanifandonlyifR/P(R)isunitregular.

ProofOnedirectionisobvious.

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Theorem11Everyabelianπ-regularringisunitJ-cleanring.

ProofIn[3,Lemma5]weknowthatinabelianπ-regularringNil(R)=J(R)andbyCorollary1wegetthateveryelementxinitcanbewriteintheformofx=eu+wwheree∈Id(R), u∈U(R)andw∈Nil(R),thenwegettheresult.

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3 Related Rings

Inthissection,wefurtherconsiderunitJ-cleannessforvariousrelatedrings.

Theorem12 ∏RiisafiniteunitJ-cleanifandonlyifeveryRiisunitJ-clean.

ProofOnedirectionisobvioussinceeveryRiisthehomomorphicimageof∏Ri.

OntheotherhandifeveryRiisunitJ-clean,thenforanyelement(a1,a2,a3……an)∈∏Riforsomen∈Z,thenai=ri+jiwhereri∈Ureg(Ri)andji∈J(Ri),thus(a1,a2,a3……an)=(r1,r2,r3……rn)+(j1,j2,j3……jn)where(r1,r2,r3……rn)∈Ureg(∏Ri)and(j1,j2,j3……jn)∈J(∏Ri),so∏RiisunitJ-clean.

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Theorem13 R[[x]]isunitJ-cleanifandonlyifRisunitJ-clean.

ProofOnedirectionisobvioussinceRisthehomomorphicimageofR[[x]].

OntheotherhandifRisunitJ-clean,thenforanyelementa0+a1x+a2x2+……∈R[[x]],thena0=r0+j0wherer0∈Ureg(R)andj0∈J(R),thus(a0+a1x+a2x2+……=r0+j0+a1x,a2x2……where(r0∈Ureg(R[[x]])andj0+a1x+a2x2+……∈J(R[[x]]),sinceJ(R[[x]])havetheformofj0+a1x+a2x2+……wherej0∈J(R),soR[[x]]isunitJ-clean.

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SetR×M={(r,m)|r∈R,m∈RMR}wedefinetheoperationby(r,m)+(s,v)=(r+s,m+v), (r,m)(s,v)=(rs,rv+ms).ThenR×Mformsaring,whichiscalledthetrivialextensionsofRandM.AsweallknowJ(R×M)={(r,m)|r∈J(R),m∈RMR}.

Theorem16LetRbearing.ThenRisunitJ-cleanifandonlyifR×M={(r,m)|r∈R,m∈RMR}isunitJ-clean.

Proof IfRis unitJ-clean, for any (r,m)∈R×MsinceRis unitJ-clean,r=eu+jwheree∈Id(R),u∈U(R),j∈J(R), we have (r,m)=(eu,0)+(j,m) where (eu,0)∈Ureg(R×M) and (j,m)∈J(R×M) which impliesR×Mis unitJ-clean whenRis unitJ-clean.

Conversely, ifR×Mis unitJ-clean, we setP=(0,M), then we haveR?R×M/PsoRis unitJ-clean whenR×Mis unitJ-clean.

:

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[2] VAMOS P. 2-good rings[J]. The Quarterly Journal of Mathematics,2005,56(3):417-430.

[3] BADAWI A. On abelian π-regular rings[J]. Communication in Algebra,1997,25(4):1009-1021.

[4] NICHOLSON W K, ZHOU Y. Clean general rings[J]. J Algebra,2005,291(1):297-311.

[5] WANG Y, REN Y L. 2-good rings and their extentions[J]. Bull Korean Math Soc,2013,50(5):1711-1723.

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關(guān)于UnitJ-clean環(huán)

沈洪地 ，陳煥艮

(杭州師范大學(xué)理學(xué)院，浙江 杭州310036)

一個(gè)元素叫做右單位J-clean(左單位J-clean)如果在R中存在一個(gè)單位u，使得au(ua)是J-clean 的.一個(gè)環(huán)R叫做右單位J-clean(左單位J-clean)環(huán)當(dāng)且僅當(dāng)環(huán)中的每個(gè)元素都是右單位J-clean(左單位J-clean)的.文章得到了以下幾個(gè)結(jié)論：每個(gè)J-clean 環(huán)是 unitJ-clean 環(huán), 每個(gè)unitJ-clean 環(huán)是 unit clean環(huán)，每個(gè)2-good 環(huán)是unit clean 環(huán)，但是以上三個(gè)結(jié)論反過來就不正確.文章還證明了一個(gè)unitJ-clean 環(huán)，那么它是2-good 環(huán)當(dāng)且僅當(dāng)1能表示成兩個(gè)單位的和.當(dāng)R是一個(gè)阿貝爾環(huán)，I是一個(gè)R的包含在Jacobson根里的理想，那么R是unitJ-clean 環(huán)當(dāng)且僅當(dāng)(1)R/I是unitJ-clean 的.(2)冪等元關(guān)于J(R)可提升.

J-clean環(huán)；unitJ-clean環(huán)；冪等元；Jacobson根

date：2015-06-16

Supported by the Natural Science Foundation of Zhejiang Province(LY13A010019).

CHEN Huanyin (1963—)，Male，Professor，ph. Doctor, majored in algebra of basic mathematics. E-mail：huanyinchen@aliyun.com

10.3969/j.issn.1674-232X.2016.02.009

O153.3 MSC2010: 16E50,16S34,16U10 Article character: A

1674-232X(2015)02-0163-08