XIANG YUE-MINGAND OUYANG LUN-QUN

(1.School of Mathematics and Computational Science,Huaihua University,Huaihua,Hunan,418000)

(2.Department of Mathematics,Hunan University of Science and Technology,Xiangtan,Hunan,411201)

Abstract:Let R be a ring and J(R)the Jacobson radical.An element a of R is called(strongly)J-clean if there is an idempotent e∈R and w∈J(R)such that a=e+w(and ew=we).The ring R is called a(strongly)J-clean ring provided that every one of its elements is(strongly)J-clean.We discuss,in the present paper,some properties of J-clean rings and strongly J-clean rings.Moreover,we investigate J-cleanness and strongly J-cleanness of generalized matrix rings.Some known results are also extended.

Key words:J-clean ring,strongly J-clean ring,generalized matrix ring

1 Introduction

Throughout this paper R is an associative ring with identity and all modules are unitary.We denote the Jacobson radical and the unit group of R by J(R)and U(R),respectively.We use Mn(R)to stand for the ring of n×n matrices over a ring R.

An element a of a ring R is(strongly)clean provided that a is the sum of an idempotent e and a unit u in R(such that e and u commute).A ring R is(strongly)clean provided that every element in R is(strongly)clean.Clean rings were first de fined by Nicholson(see[1])as a class of exchange rings.It is well known that unit regular rings and semiperfect rings are also clean rings.The class of strongly clean rings was introduced in[2].It was shown that all strongly π-regular rings are strongly clean and that a strongly clean endomorphism satis fies a generalized version of Fitting’s lemma.In recent decades,many researchers studied such rings from various different perspectives(see[3]–[9]).Among the others,Chen[4]developed the concept of strongly J-clean rings to construct a subclass of strongly clean rings which have stable range one.Here an element a∈R is strongly J-clean provided that there exists an idempotent e and an element w∈J(R)such that a=e+w and ew=we.The ring R is strongly J-clean if every element is strongly J-clean.It was proven that the ring of all 2×2 matrices over a commutative local ring is not strongly J-clean.

2 (Strongly)J-clean Rings

For convenience,we use the following de finition in the sequel.

De finition 2.1Let R be a ring.An element a of R is said to be J-clean if there is an idempotent e∈R and w∈J(R)such that a=e+w.The ring R is called a J-clean ring provided that every one of its elements is J-clean.The concept of J-clean general rings(with or without identity)is consistent with that of semiboolean rings in[11].

Remark 2.1(1) Idempotents or elements in J(R)are J-clean elements,and then boolean rings are J-clean rings.

(3)Obviously,the homomorphism image of a J-clean ring is J-clean.

Lemma 2.1Let R be a ring and a∈R.Then a is J-clean in R if and only if there exists an idempotent e∈R and a unit u∈R such that ua=eu+1 and u?1?(1?2e)∈J(R).Proof. Suppose that a∈R is J-clean.Then we have

with f2=f∈R and w∈J(R).Writing e′=1?f and v=2f?1+w.Then

where e′2=e′∈ R and v ∈ U(R).Multiplying by v?1=u from the left,we get

On the other hand,

It implies that

Conversely,suppose that there exists an idempotent e∈R and a unit u∈R such that ua=eu+1 and u?1?(1?2e)∈ J(R).Then we have

where u?1(1? e)u is an idempotent and u?1(u?1? (1? 2e))u ∈ J(R).Therefore,a is J-clean in R.The proof is completed.

By Lemma 2.3 and Theorem 2.1 of[6]we have

Corollary 2.1Every J-clean element is clean.

In[4],Chen proven that R is a strongly J-clean ring if and only if R/J(R)is a boolean ring and idempotents strongly lift modulo J(R)if and only if R/J(R)is a boolean ring and R is strongly clean.Analogously,we have the next results for J-clean rings.

Theorem 2.1Let R be a ring.Then the following are equivalent:

(1)R is a J-clean ring;

(2)R/J(R)is a boolean ring and idempotents lift modulo J(R);

(3)R/J(R)is a boolean ring and R is clean.

Let R be a ring and M a bimodule over R.The trivial extension of R and M is R∝M={(x,m)|x∈R,m∈M}with addition de fined componentwise and multiplication de fined by(x,m)(y,n)=(xy,xn+my).In view of Theorem 2.1 and Lemma 3.10 of[12],it is easy to verify that the trivial extension R∝M is J-clean if and only if R is J-clean.

Proposition 2.1Let R be a J-clean ring.Then

(1)2∈J(R);

(2)Every nilpotent element of R is contained in J(R);

(4)For any a,b∈R,if 1?ab∈J(R),then 1?ba∈J(R).

Proof.(1)Clearly,there exists an idempotent e∈R and a w∈J(R)such that 2=e+w.Hence,1?e=w?1∈U(R),and then e=0.As a result,2=w∈J(R).

(2)For any a∈N(R),a=e+w,where e2=e∈R and w∈J(R).Then

It implies that 1?e=1 and hence e=0.Therefore,a=w∈J(R).

(3)Assume that ab=1.Since R is J-clean,there exists e2=e and a w∈J(R)such that

Thus

It implies that e=ewb∈J(R),and hence e=0.Then 1+a=w∈J(R),and so a∈U(R).

(4)If 1?ab∈J(R),then ab∈U(R).By(3),b∈U(R).So

Proposition 2.2Let R be a ring.Then the following are equivalent:

(1)R is a boolean ring;

(2)R is a J-clean ring and J(R)=0.

Lemma 2.2An element a∈R is strongly J-clean if and only if a is strongly clean and a?a2∈J(R).

Proof.If a is strongly J-clean,then a=e+w,where e2=e,w∈J(R)and ew=we.Thus

So

Conversely,let a=e+u be a strongly clean decomposition in R and a?a2∈J(R).Then

and

It follows that 1?2e?u∈J(R).So a=(1?e)+(2e?1+u)is a strongly J-clean decomposition in R.

Corollary 2.2A unit u of a ring R is strongly J-clean if and only if 1?u∈J(R).

For a∈R,

and

An element a∈R is called quasipolar(see[13])if there existssuch that a+p∈U(R)and ap∈Rqnil.A ring is quasipolar if and only if its elements are quasipolar.A local ring is quasipolar but need not be strongly J-clean.However,we have the following result.

Proposition 2.3If R is a quasipolar ring and for each unit u∈R,1?u∈J(R),then R is strongly J-clean.

Proof.For each a∈R,since R is quasipolar,a=p+u,where p2=p∈comm2(a),u∈U(R)and ap∈Rqnil.Now it is enough to show that the element 2p+u is a unit in R.In fact,considering the Peirce decomposition with respect to the idempotent p,we have

and

By hypothesis,2p+u?1∈J(R),and hence a=(1?p)+(2p+u?1)is a strongly J-clean decomposition.The proof is completed.

An element r∈R is called strongly nil clean(see[5])if there is an idempotent e∈R and a nilpotent b∈R such that r=e+b and eb=be.A ring is called strongly nil clean if every one of its elements is strongly nil clean.

Proposition 2.4Every strongly nil clean ring is strongly J-clean.

Proof. If R is strongly nil clean,in view of Theorem 2.7 of[7],then R/J(R)is boolean.So the result follows from Theorem 2.3 of[4]since R is also strongly clean.

Example 2.2Z2[[x]]is a strongly J-clean ring but not strongly nil clean.

3 (Strongly)J-cleanness of Generalized Matrix Rings

Proof. By Theorem 2.1,T/J(T)is boolean.By Proposition 2.6 of[8],

and hence it must be that M/M0=0 and N/N0=0,i.e.,M=M0and N=N0.It implies that MN ? J(A)and NM ? J(B).Thus,and B/J(B)are also boolean.

On the other hand,T is also clean.Then A and B are clean by Theorem 3.3 of[8].Therefore,A and B are J-clean by Theorem 2.1 again.Conversely,T is clean by Theorem 3.3 of[8].T/J(T)is boolean in view of Lemma 3.3 of[7].So T is J-clean by Theorem 2.1.

Corollary 3.1Let R be a ring with s∈C(R).Then Ks(R)is J-clean if and only if R is J-clean and s∈J(R).

Corollary 3.2Let R be a ring.The upper triangular matrix Tn(R)is J-clean for some n∈N+if and only if R is J-clean.

Corollary 3.3The matrix ring Mn(R)(n≥2)is never(strongly)J-clean.

Now we are in position to consider when a single matrix is strongly J-clean in Ks(R).The following lemmas is due to[9].

Lemma 3.1Let R be a ring with s∈C(R)and let J=J(R).Then

Lemma 3.2Let E2=E∈Ks(R).If E is equivalent to a diagonal matrix in Ks(R),then E is similar to an idempotent diagonal matrix in Ks(R).

Let a,b∈R.We use the notation a～b to mean that a is similar to b,that is,b=u?1au for some u∈U(R).It is easy to verify that a is strongly J-clean if and only if b is strongly J-clean.

Theorem 3.2Let R be a local ring with s∈C(R)and A∈Ks(R).Then the following statements are equivalent:

(1)A is strongly J-clean in Ks(R);

Proof. (2)?(1).If either A or I2?A∈J(Ks(R)),then A is strongly J-clean in Ks(R).Now for any w1,w2∈J(R),

is strongly J-clean in Ks(R),so is A.

Case 1.If a/∈J(R),then a∈U(R),and

Case 2.If b/∈J(R),then b∈U(R),and

Case 3.If a∈J(R)and b∈J(R).Noting thatwe haveby Lemma 3.1.Thus,s∈U(R),and henceor.Assume thatThen x∈U(R).SinceOn the other hand,

which is equivalent to a diagonal matrix as Case 1(sx∈U(R)).

Now by Lemma 3.2,there exists a unit U of Ks(R)such that

where f1,f2are idempotents in R.As E is a non-trivial idempotent of Ks(R),we see that f1=1,f2=0 or f1=0,f2=1.Since UAU?1=UEU?1+UWU?1is also strongly J-clean in Ks(R),

Write V=(vij)=UWU?1∈ J(Ks(R)).It follows from EW=WE that

The next result was proved in Theorem 5.2 of[4].

Theorem 3.3Let R be a commutative local ring and let A∈Ks(R).Then the following statements are equivalent:

(1)A is strongly J-clean in Ks(R);

(2)A∈J(Ks(R))or I2?A∈J(Ks(R))or the equation

has a root in J(R)and a root in 1+J(R).

Proof.(1)?(2).Suppose that A is strongly J-clean in Ks(R)andandIn view of Theorem 3.2,A is similar to

where w1,w2∈J(R).By Lemma 14 of[9],

and

Thus,1+w1and w2are also roots of

(2)?(1).If A∈J(Ks(R))or I2?A∈J(Ks(R)),then A is strongly J-clean in Ks(R).Now we assume that the equation

has a root λ1∈ 1+J(R)and a root λ2∈ J(R),and write Then

and hence

By Lemma 14 of[9],A/∈U(Ks(R))and I2?A/∈U(Ks(R)).Now we discuss in two cases.

and hence

has a root λ1∈ 1+J(R)and a root λ2∈ J(R).It is easy to see that

So

Moreover,we can check that P∈U(Ks(R))and

Assume

By Lemma 1 of[9],it is enough to show that

is strongly J-clean in Ks(R).So set

and we can check that Q∈U(Ks(R))and

Therefore,A is strongly J-clean by Theorem 3.2.

Case 2.Let s∈U(R).We first prove that A is similar to

where u∈1+J(R),λ∈J(R).

Case 2.1.If y∈U(R),then

Let

If λ∈U(R),then

If 1?μ∈U(R),then

which contradict with I2?A/∈U(Ks(R)).So 1?μ∈J(R),and henceμ∈1+J(R).

Case 2.2.If x∈U(R),then

Case 2.3.Assume that x∈J(R)and y∈J(R).Then we claim that a∈J(R)and b∈U(R),or a∈U(R)and b∈J(R).

Otherwise,if a,b∈U(R),then

If a,b∈J(R),then

and hence we back to Case 2.1.

From the above discussion,

So

has a root λ1∈ 1+J(R)and a root λ2∈ J(R).Thus one can set

Noting that dets(X)=λ1?λ2∈U(R).So X ∈U(Ks(R))and we can check that

Therefore,A is strongly J-clean by Theorem 3.2 again.

So Theorem 6.3 of[4]can be developed as a corollary.

Corollary 3.5Let R be a commutative local ring.Then the following statements are equivalent:

(1)A∈M2(R)is strongly J-clean;

(2)A∈J(M2(R))or I2?A∈J(M2(R))or the equation

has a root in J(R)and a root in 1+J(R).