Unit 1 stable range for ideals

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PII. S0161171204402361 http://ijmms.hindawi.com © Hindawi Publishing Corp.

UNIT

1 -STABLE RANGE FOR IDEALS

HUANYIN CHEN and MIAOSEN CHEN

Received 12 February 2004

We investigate necessary and sufficient conditions under which a ring satisfies unit 1-stable range for an ideal. As an application, we prove thatRsatisfies unit 1-stable range forIif and only if QM2(R)satisfies unit 1-stable range for QM2(I).

2000 Mathematics Subject Classiﬁcation: 16U99, 16E50.

LetR be a ring with identity 1. We say thatR satisﬁes unit 1-stable range in case

ax+b=1 witha,x,b∈R implying thata+bu∈U(R). Many authors studied unit 1-stable range such as those of [1,2,3,4,5,6]. Following the authors, a ringRsatisﬁes unit 1-stable range for an idealIprovided thatax+b=1 witha∈I, x,b∈Rimplying thatx+ub∈U(R)for some unitu∈U(R). LetR=Z/2Z⊕Z/3ZandI=0⊕Z/3Z. Then

Rsatisfies unit 1-stable range forI, while in factRdoes not satisfy unit 1-stable range. Thus the concept of unit 1-stable range for an ideal is a nontrivial generalization of that of rings satisfying such stable range condition. In this note, we investigate necessary and sufficient conditions under which a ringRsatisfies unit 1-stable range for an ideal. It is shown thatRsatisfies unit 1-stable range forIif and only if QM2(R)satisfies unit 1-stable range for QM2(I).

Throughout, all rings are associative with identity.Mn(R)denotes the ring ofn×n matrices overR, GLn(R)denotes then-dimensional general linear group ofR, TMn(R) denotes the ring of alln×nlower triangular matrices overR, and TMn(I)denotes the ideal of alln×nlower triangular matrices overI. Clearly, TMn(I)is an ideal of TMn(R). We begin with the following.

Lemma1. _Let_I_{be an ideal of a ring}_R_{. Then the following are equivalent:}

(1) Rsatisfies unit1-stable range forI;

(2) for anyx∈R,y∈I, there existsa∈Rsuch that1+xa,y+a∈U(R).

Proof. ₍₁₎⇒_{(2). Given any}_x∈_R_,_y∈_I_{, we have}_yx+₍₁−_yx)=_{1; hence, there}

existsu∈U(R)such thatx+u(1−yx)∈U(R), so 1+(u−1₋_y)x_∈_U(R)_{. Set} _a₌ u−1_−y_{. Then 1}_+ax,y_+a_∈_U(R)_{. Clearly, 1}_+ax_∈_U(R)_{if and only if 1}_+xa_∈_U(R)_. Therefore, 1+xa,y+a∈U(R).

(2)⇒(1). Suppose thatax+b=1 witha∈I, x,b∈R. Then we haver ∈R such thata+r ,1+(−x)r∈U(R). Setu=a+r. We get 1−x(a−u)=1+(−x)r∈U(R); hence, 1−(a−u)x∈U(R). This infers thatb+ux∈U(R), and sox+u−1_b_∈_U(R)_{, as} asserted.

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Theorem2. IfRsatisfies unit1-stable range forI, then so doesM_n(R)forM_n(I).

Proof. Assume thata1N1 M1B1

∈Mn(I),Ma22NB22

∈Mn(R)witha1∈I, a2∈R, B1∈ Mn−1(I), B2∈Mn−1(R), N1 ∈M1×(n−1)(I), N2∈ M1×(n−1)(R), M1∈ M(n−1)×1(I), and M2∈M(n−1)×1(R). UsingLemma 1, we can choosea∈Rsuch thata1+a=u1∈U(R), 1+a2a=v1∈U(R). Assume that Mn−1(R)satisﬁes unit 1-stable range forMn−1(I). Clearly,B1−M1u−11N1∈Mn−1(I). So we haveB∈Mn−1(R)such that(B1−M1u−11N1)+ B=U2∈GLn−1(R)andIn−1+(−M2v1−1N2+B2)B=V2∈GLn−1(R). Hence,

a1 N1 M1 B1 + a 0 0 B =

u1 N1 M1 B1+B

,

In+

a2 N2 M2 B2

a 0 0 B =

v1 N2B M2a In−1+B2B

.

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We check that

u1 N1 M1 B1+B

−1 =



u−11+u−11N1U2−1M1u−11 −u−11N1U2−1 −U−1

2 M1u−11 U2−1  _,

v1 N2B M2a In−1+B2B

−1 =



v1−1+v1−1N2BV2−1M2av1−1 −v1−1N2BV2−1 −V−1

2 M2av1−1 V2−1  _.

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By induction andLemma 1,Mn(R)satisﬁes unit 1-stable range forMn(I).

Corollary3. IfRsatisfies unit1-stable range, then so doesM_n(R)for alln∈N.

Proof. By choosingI=RinTheorem 2, we complete the proof.

Theorem4. LetIbe an ideal of a ringR. Then the following are equivalent:

(2) TMn(R)satisfies unit1-stable range forTMn(I).

Proof. It suﬃces to show that the result holds forn=2. Suppose that R

satis-ﬁes unit 1-stable range forI. Assume that a1 0 m1b1

∈TM2(I), a2 0

m2b2

∈TM2(R) with a1,b1,m1∈I,a2,b2,m2∈R. UsingLemma 1, we can choosea∈Rsuch thata1+a= u1∈U(R), 1+a2a=v1∈U(R)and we haveb∈R such thatb1+b=u2∈U(R)and 1+b2b=v2∈U(R). Hence,

a1 0 m1 b1

+ a 0 0 b =

u1 0 m1 b1+b

,

I2+

a2 0 m2 b2

a 0 0 b =

v1 0

m2a I2+b2b

.

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Analogously toTheorem 2, we have

u1 0 m1 b1+b

,

v1 0

m2a I2+b2b

∈GL2

TM2(R)

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ByLemma 1again, TM2(I)is unit 1-stable.

We now establish the converse. Given anyx∈R,y∈I, we have diag(x,1)∈TM2(R) and diag(y,0)∈TM2(I). So there exists

_a0 b c

∈TM2(R)such that diag(1,1)+diag(x,1) _a0

b c

,

diag(y,0)+a0 b c

∈GL2(TM2(R)). Therefore, 1+xa,y+a∈U(R), as required.

LetIbe an ideal of a unital complexC∗-algebraR. ByTheorem 4and [3, Corollary 6], we prove that if every element ofIis a sum of a unitary and a unit, then every square lower-triangular matrix overI is a sum of two invertible matrices. LetIbe an ideal of a ringR. Deﬁne QM2(R)= a b_{c d}

|a+c=b+d, a,b,c,d∈Rand QM2(I)= a b_{c d}

| a+c=b+d, a,b,c,d∈I. It is easy to verify that QM2(I)is an ideal of QM2(R).

Corollary5. LetIbe an ideal of a ringR. Then the following are equivalent:

(2) QM2(R)satisfies unit1-stable forQM2(I).

Proof. (1)⇒(2). We construct a mapψ: QM₂(R)→TM₂(R)given bya b c d

→a+c 0 c d−c

fora b c d

∈QM2(R). For any _x0

z y

∈TM2(R), we haveψ

x−z x−y−z z y+z

=x0

z y

. Thus we prove thatψis a ring isomorphism. Alsoψ|QM2(I): QM2(I)TM2(I). Thus we complete the proof byTheorem 4.

As an immediate consequence ofCorollary 5, we prove thatRsatisﬁes unit 1-stable range if and only if so does QM2(R). This result gives a new kind of rings satisfying unit 1-stable range.

Theorem6. LetIbe an ideal of a ringR. Then the following are equivalent:

(2) there exists a complete set of idempotents{e1,...,en}such that all eiReisatisfy

unit1-stable range foreiIei.

Proof. ₍₁₎⇒_{(2) is clear.}

(2)⇒(1). One easily checks that

I      

e1Ie1 ··· e1Ien ..

. . .. ...

enIe1 ··· enIen

     , R      

e1Re1 ··· e1Ren ..

. . .. ...

enRe1 ··· enRen

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By induction, it suﬃces to prove that the result holds forn=2. Assume thata1 n1 m1b1

∈ e1Ie1e1Ie2

e2Ie1e2Ie2

,ma22nb22

∈e1Re1e1Re2 e2Re1e2Re2

. According to Lemma 1, we can choose a∈e1Re1 such thata1+a=u1∈U(e1Re1),e1+a2a=v1∈U(e1Re1). Also we haveb∈e2Re2 such that (b1−m1u−11n1)+b=u2∈U(e2Re2)and e2+(−m2v1−1n2+b2)b=v2∈ U(e2Re2). Hence,



a1 n1 m1 b1  ₊

 a 0

0 b

 ₌



u1 n1 m1 b1+b

 _,

diage1,e2

+ 

a2 n2 m2 b2    a 0

0 b

 ₌



 v1 n2b m2a e2+b2b

 _.

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Similarly toTheorem 2, we show that



u1 n1 m1 b1+b

 _,



 v1 n2b m2a e2+b2b

 _∈_U



e1Re1 e1Re2 e2Re1 e2Re2 

_. (7)

It follows byLemma 1thate1Ie1e1Ie2 e2Ie1e2Ie2

satisﬁes unit 1-stable range fore1Ie1e1Ie2 e2Ie1e2Ie2

, as required.

Corollary7. LetRbe a ring. Then the following are equivalent:

(1) Rsatisfies unit1-stable range;

(2) there exists a complete set of idempotents{e1,...,en}such that all eiReisatisfy

unit1-stable range.

Proof. It is an immediate consequence ofTheorem 8.

Recall thatI is a regular ideal ofR. In case for anyx∈I, there existsy∈R such thatx=xyx. We prove that unit 1-stable range condition for an ideal is right and left symmetric for regular ideals.

Theorem8. LetIbe a regular ideal of a ringR. Then the following are equivalent:

(1) Rsatisfies unit1-stable forI;

(2) for anyx,y∈I, there exists somea∈U(R)such that1+xa,y+a∈U(R);

(3) ax+b =1 with a∈I, b∈R implying that there exists u∈U(R) such that a+bu∈U(R).

Proof. ₍₁₎⇒_{(2) is trivial by}_{Lemma 1.}

(2)⇒(3). Suppose thatax+b=1 witha∈I, b∈R. By the regularity ofI, we have a

c∈Rsuch thata=aca. Hence,a(cax)+b=1. Sincecax∈I, it follows byLemma 1 that there exists ad∈Rsuch that 1+ad,cax+d∈U(R). Setu=cax+d. Then we have 1+ad=1+a(u−cax)=au+b∈U(R). Therefore,a+bu−1_∈_U(R)_.

(3)⇒(2). For any x,y∈I, we havexy+(1−xy)=1. By hypothesis, we can ﬁnd

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Corollary9. LetIbe a regular ideal of a ringR. Then the following are equivalent:

(2) Rop_{satisfies unit}₁_{-stable range for}_Iop_.

Proof. ₍₁₎⇒_{(2). For any}_xop_,yop∈_Iop_{, we have}_x,y∈_R_{. Hence, there exists some}

a∈U(R)such that 1+xa,y+a∈U(R)byTheorem 8. Clearly, 1+xa∈U(R)if and only if 1+ax∈U(R). So 1op_+xop_aop_,xop_+aop_∈_U(Rop₎_{. Therefore,}_Iop_{is unit 1-stable} byTheorem 8again.

(2)⇒(1) is proved in the same manner.

Recall that an idealIof a ringRhas stable rank one in caseaR+bR=Rwitha∈1+I,

b∈Rimplying that there existsy∈Rsuch thata+by∈U(R). It is well known that an idealIof a regular ringRhas stable range one if and only ifeReis unit-regular for all idempotentse∈I.

Theorem10. LetIbe an ideal of a regular ringR. IfIhas stable range one, then the

following are equivalent:

(2) ife∈I,f∈1+Iare idempotents such thateR+f R=R, then there existu,v∈ U(R)such thateu+f v=1.

Proof. ₍₁₎⇒_{(2). Suppose that}_eR+_{f R}=_R_{with idempotents}_e∈_I_,_f∈₁+I_{. Since}_I

has stable rank one, we can ﬁnd ay∈Rsuch thatey+f=u∈U(R). Hence,eyu−1₊ f u−1₌_{1. As} _R _{satisﬁes unit 1-stable range for}_I_{, there exists}_v _∈_U(R) _{such that} e+f u−1_v₌_w_∈_U(R)_by_{Theorem 8. Hence,}_ew−1_{+f u}−1_vw−1₌_{1, as required.}

(2)⇒(1). Givenax+b=1 with a∈I, x,b∈R, thenb=1−ax∈1+I. Since R is regular, we havec ∈R such that b=bcb. Clearly,c ∈1+I. As I has stable range one, it follows frombc+(1−bc)=1 that b+(1−bc)y ∈U(R)for ay∈R. Hence,

b+(1−bc)y=u, and sob=bcb=bcu. Similarly, we haved∈Rsuch thata=ada. Sincea+(1−ad)d+(1−ad)(1−d)=1 witha+(1−ad)∈1+I, we can ﬁndz∈R

such thata+(1−ad)+z(1−ad)(1−d)=v∈U(R). This shows thata=ada=adv. Lete=adandf=bc. Thene∈Iandf∈1+I. Clearly,eR+f R=R. By hypothesis, there are s,t∈U(R)such thates+f t =1. Therefore,av−1_s₊_bu−1_t₌_{1, and then} a+bu−1_ts−1_v_∈_U(R)_{. According to}_{Theorem 8, we obtain the result.}

LetIbe a bounded ideal of a regular ringR. As a result, we deduce thatRsatisﬁes unit 1-stable range forIif and only ife∈I,f∈1+Iare idempotents such thateR+f R=R; then there existu,v∈U(R)such thateu+f v=1.

Corollary11. LetRbe unit-regular. Then the following are equivalent:

(1) Rsatisfies unit1-stable range;

(2) ife,f ∈Rare idempotents such thateR+f R=R, then there existu,v∈U(R) such thateu+f v=1.

Proof. It is clear byTheorem 10.

Acknowledgment. This work was supported by the Natural Science Foundation

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[2] ,Exchange rings satisfying unit 1-stable range, Kyushu J. Math.54(2000), no. 1, 1–6. [3] H. Chen and M. Chen,On unit 1-stable range, J. Appl. Algebra Discrete Struct.1(2003), no. 3,

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Huanyin Chen: Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China

E-mail address:[email protected]

Miaosen Chen: Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China