PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 100(114) (2016), 287–298 DOI: 10.2298/PIM1614287C

DECOMPOSITIONS OF 2 × 2 MATRICES OVER LOCAL RINGS

Huanyin Chen, Sait Halicioglu, and Handan Kose

Abstract. An element a of a ring R is called perfectly clean if there exists an idempotent e ∈ comm^{2}(a) such that a − e ∈ U(R). A ring R is perfectly clean in case every element in R is perfectly clean. In this paper, we completely determine when every 2 × 2 matrix and triangular matrix over local rings are perfectly clean. These give more explicit characterizations of strongly clean matrices over local rings. We also obtain several criteria for a triangular matrix to be perfectly J-clean. For instance, it is proved that for a commutative local ring R, every triangular matrix is perfectly J-clean in T_{n} (R) if and only if R is strongly J-clean.

1. Introduction

The commutant and double commutant of an element a in a ring R are defined by comm(a) = {x ∈ R | xa = ax}, comm^{2}(a) = {x ∈ R | xy = yx for all y ∈ comm(a)}, respectively. An element a ∈ R is strongly clean provided that there exists an idempotent e ∈ comm(a) such that a − e ∈ U(R). A ring R is called strongly clean in the case that every element in R is strongly clean. Strongly clean matrix rings and triangular matrix rings over local rings have been extensively studied by many authors (cf. [1, 2, 5, 6] and [12, 13]. An element a ∈ R is quasipolar provided that there exists an idempotent e ∈ comm^{2}(a) such that a+e ∈ U(R) and ae ∈ R^{qnil} , where R^{qnil} = {x ∈ R | 1 + xr ∈ U(R) for any r ∈ comm(x)}. A ring R is called quasipolar if every element in R is quasipolar. As is well known, a ring R is quasipolar if and only if for any a ∈ R there exists a b ∈ comm^{2}(a) such that b = bab and b − b^{2}a ∈ R^{qnil} . This concept has evolved from Banach algebra. In fact, for a Banach algebra R,

a ∈ R^{qnil} ⇔ lim_{n→∞} ∥a^{n}∥^{\frac{1}{n}} = 0.

It is shown that every quasipolar ring is strongly clean. Recently, quasipolar 2 × 2 matrix rings and triangular matrix rings over local rings were also studied from different point of views (cf. [7, 9, 11]).

2010 Mathematics Subject Classification: 16S50; 16S70; 16U99.

Key words and phrases: perfectly clean ring; perfectly J-clean ring; quasipolar ring; matrix; triangular matrix.

Communicated by Žarko Mijajlović.

The motivation for this article is to introduce a medium class between strongly clean rings and quasipolar rings, and then explore more explicit decompositions of 2 × 2 matrices over a local ring. An element a of a ring R is called perfectly clean if there exists an idempotent e ∈ comm^{2}(a) such that a − e ∈ U(R). A ring R is perfectly clean in the case every element in R is perfectly clean. We completely determine when every 2×2 matrix and triangular matrix over local rings are perfectly clean. These also give more explicit characterizations of strong clean matrices over local rings, and enhance many known results, e.g., [5, Theorem 8], [11, Theorem 2.8] and [12, Theorem 7]. Replaced U(R) by J(R), we introduce perfectly J-clean rings as a subclass of perfectly clean rings. Furthermore, we show that strong J-cleanness for triangular matrices over a local ring can be enhanced to such stronger properties. These also generalize the corresponding properties of J-quasipolarity, e.g., [8, Theorem 4.9].

We write U(R) and J(R) for the set of all invertible elements and the Jacobson radical of R; M_{n}(R) and T_{n}(R) stand for the rings of all n × n matrices and triangular matrices over a ring R.

2. Perfect rings

Clearly, an abelian exchange ring is perfectly clean. Every quasipolar ring is perfectly clean. For instance, every strongly π-regular ring. In fact, we have {quasipolar rings} ⊊ {perfectly clean rings} ⊊ {strongly clean rings}. In this section, we explore the properties of perfect rings, which will be used in the sequel. We begin with

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

(1) R is perfectly clean.

(2) For any a ∈ R, there exists an x ∈ comm^{2}(a) such that x = xax and 1 − x ∈ (1 − a)R ∩ R(1 − a).

Proof. (1) ⇒ (2) For any a ∈ R, there exists an idempotent e ∈ comm^{2}(a) such that u := a − e ∈ U(R). Set x = u^{−1} (1 − e). Let y ∈ comm(a). Then ay = ya. As uy = (a − e)y = y(a − e) = yu, we get u^{−1} y = yu^{−1} . Thus, xy = u^{−1} (1 − e)y = u^{−1} y(1 − e) = yu^{−1} (1 − e) = yx. This implies that x ∈ comm^{2}(a). Further, xax = u^{−1} (1 − e)(u + e)u^{−1} (1 − e) = u^{−1} (1 − e) = x. Clearly, u = (1 − e) − (1 − a), and so 1 − u^{−1} (1 − e) = u^{−1} (1 − a). This implies that 1 − x ∈ R(1 − a). Likewise, 1 − x ∈ (1 − a)R as (1 − e)u^{−1} = u^{−1} (1 − e). Therefore 1 − x ∈ (1 − a)R ∩ R(1 − a), as required.

(2) ⇒ (1) For any a ∈ R, there exists an x ∈ comm^{2}(a) such that x = xax and 1 − x ∈ (1 − a)R ∩ R(1 − a). Write e = 1 − ax. If y ∈ comm(a), then ay = ya, and so axy = ayx = yax. This shows that ey = ye; hence, e ∈ comm^{2}(a). In addition, ex = xe = 0. Write 1 − x = (1 − a)s = t(1 − a) for some s, t ∈ R. Then

(a − e)(x − es) = ax − aes + es = ax + (1 − a)es
= ax + e(1 − a)s = ax + e(1 − x) = ax + e = 1.

Likewise, (x − te)(a − e) = 1. Therefore a − e ∈ U(R), as desired. □

DECOMPOSITIONS OF 2 × 2 MATRICES OVER LOCAL RINGS 289

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

(1) R is perfectly clean.

(2) For any a ∈ R, there exists an idempotent e ∈ comm^{2}(a) such that eae ∈ U(eRe) and (1 − e)(1 − a)(1 − e) ∈ U((1 − e)R(1 − e)).

Proof. (1) ⇒ (2) For any a ∈ R, it follows from Theorem 2.1 that there exists an x ∈ comm^{2}(a) such that x = xax and 1 − x ∈ (1 − a)R ∩ R(1 − a). Write 1 − x = (1 − a)s = t(1 − a) for some s, t ∈ R. Set e = ax. For any y ∈ comm(a), we have ay = ya, and so ey = (ax)y = a(yx) = (ay)x = y(ax) = ye. Hence, e^{2} = e ∈ comm^{2}(a). Clearly, (eae)(exe) = (exe)(eae) = e; hence, eae ∈ U(eRe). Furthermore, 1−e = (1−x)+(1−a)x = (1−a)(s+x). This shows that (1−e)(1− a)(1−e)(1−x)(1−e) = 1−e. Likewise, (1−e)(1−x)(1−e)(1−e)(1−a)(1−e) = 1−e. Therefore (1 − e)(1 − a)(1 − e) ∈ U((1 − e)R(1 − e)).

(2) ⇒ (1) For any a ∈ R, we have an idempotent e ∈ comm^{2}(a) such that eae ∈ U(eRe) and (1 − e)(1 − a)(1 − e) ∈ U((1 − e)R(1 − e)). Hence, a − (1 − e) = (eae − (1 − e)(1 − a)(1 − e)) ∈ U(R). Set p = 1 − e. Then a − p ∈ U(R) with p ∈ comm^{2}(a), as desired. □

Recall that a ring R is strongly nil clean provide that every element in R is the sum of an idempotent and a nilpotent element that commutate (cf. [4] and [10]).

Theorem 2.2. Let R be a ring. Then R is strongly nil clean if and only if

(1) R is perfectly clean, (2) N(R) = {x ∈ R | 1 − x ∈ U(R)}.

Proof. Let R be strongly nil clean. For any a ∈ R, we see that a−a^{2} ∈ N(R). Write (a − a^{2})^{n} = 0. Let f(t) = ∑ _{i=0}^{n} \binom{2n}{i}t^{2n−i} (1 − t)^{i} ∈ ℤ[t]. Then we have f(t) ≡ 0 (mod t^{n}). Clearly,

f(t) + ∑_{i=n+1}^{2n} \binom{2n}{i} t^{2n−i} (1 − t)^{i} = (t + (1 − t))^{n} = 1;

hence, f(t) ≡ 1 (mod (1−t)^{n}). This shows that f(t)(1−f(t)) ≡ 0 (mod t^{n}(1−t)^{n}). Let e = f(a). Then e ∈ R is an idempotent. For any x ∈ comm(a), we see that xa = ax, and so xe = xf(a) = f(a)x = ex. Thus, e ∈ comm^{2}(a). Furthermore, a − e = a − a^{2n} + (a − a^{2})g(a) = (a − a^{2})(1 + a + a^{2} + ⋯+ a^{2n−2} + g(a)) ∈ N(R), where g(t) ∈ ℤ[t]. Thus, a = (1 − e) + (2e − 1 + a − e) with 1 − e ∈ comm^{2}(a) and 2e − 1 + a − e ∈ U(R). Therefore, R is perfectly clean.

Clearly, N(R) ⊆ {x ∈ R | 1 − x ∈ U(R)}. If 1 − x ∈ U(R), then x = e + w with e ∈ comm(x) and w ∈ N(R). Hence, 1 − e = (1 − x) + w ∈ U(R). This implies that 1 − e = 1, and so x = w ∈ N(R). Therefore N(R) = {x ∈ R | 1 − x ∈ U(R)}.

Conversely, assume that (1) and (2) hold. For any a ∈ R, there exist an idempotent e ∈ comm^{2}(a) and a unit u ∈ R such that −a = e − u. Hence, a = −e + u = (1 − e) − (1 − u). By hypothesis, 1 − u ∈ N(R). Accordingly, R is strongly nil clean. □

Corollary 2.2. Let R be a ring. Then R is strongly nil clean if and only if (1) R is quasipolar; (2) N(R) = {x ∈ R | 1 − x ∈ U(R)}.

Proof. Suppose that R is strongly nil clean. Then (2) holds by Theorem 2.2. For any a ∈ R, as in the proof of Theorem 2.2, a = e + w with e ∈ comm^{2}(a) and w ∈ N(R). Hence, a = (1−e)+(2e−1+w) where 2e−1+w ∈ U(R). Furthermore, (1 − e)a = (1 − e)w ∈ N(R) ⊆ R^{qnil} . Therefore R is quasipolar.

Conversely, assume that (1) and (2) hold. Then R is perfectly clean. Accordingly, we complete the proof by Theorem 2.2. □

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

(1) R is perfectly clean.

(2) For each a ∈ R there exists an idempotent e ∈ comm^{2}(a) such that a − e and a + e are invertible.

Proof. (1) ⇒ (2) Let a ∈ R. Then a^{2} ∈ R is perfectly clean. Thus, we can find an idempotent e ∈ comm^{2}(a^{2}) such that a^{2} − e ∈ U(R). Since a · a^{2} = a^{2} · a, we see that ae = ea. Hence, a^{2} − e = (a − e)(a + e), and therefore we conclude that a − e, a + e ∈ U(R).

(2) ⇒ (1) is trivial. □

Theorem 2.3. Let R be perfectly clean. Then for any A ∈ M_{n}(R) there exist U, V ∈ GL_{n}(R) such that 2A = U + V.

Proof. We prove the result by induction on n. For any a ∈ R, there exists an idempotent e ∈ comm^{2}(a) such that u := a − e, v := a + e ∈ U(R), by Lemma 2.1. Hence, 2a = u + v, and so the result holds for n = 1. Assume that the result holds for n ⩽ k (k ⩾ 1). Let n = k + 1, and let A ∈ M_{n}(R). Write A = ( \begin{matrix}x & α \\ β & X \end{matrix} ), where x ∈ R, α ∈ M_{1×k} (R), β ∈ M_{k×1} (R) and X ∈ M_{k}(R). In view of Lemma 2.1, we have a u ∈ U(R) such that 2x − u = v ∈ U(R). By hypothesis, we have a U ∈ GL_{k}(R) such that 2(X − 2βv^{−1} α) − U = V ∈ GL_{k}(R). Hence

2A − ( \begin{matrix}u & 0 \\ 0 & U \end{matrix}) = ( \begin{matrix}v & 2α \\ 2β & V + 4βv^{−1} α \end{matrix} ) .

It is easy to verify that

( \begin{matrix}v & 2α \\ 2β & V + 4βv^{−1} α \end{matrix} ) = \binom{1}{2βv^{−1} I_{k}} ( \begin{matrix}v & 2α \\ 0 & V \end{matrix}) ∈ GL_{n}(R).

By induction, we complete the proof. □

Corollary 2.3. Let R be a quasipolar ring. If \frac{1}{2} ∈ R, then every n×n matrix over R is the sum of two invertible matrices.

Proof. As every quasipolar ring is perfectly clean, the proof follows by Theorem 2.3. □

As a consequence, we derive the following known fact: Let R be a strongly π-regular ring with \frac{1}{2} ∈ R. Then every n × n matrix over R is the sum of two invertible matrices.

3. Matrices and triangular matrices

Recall that a ring R is local if it has only one maximal right ideal. A ring R is local if and only if for any a ∈ R either a or 1 − a is invertible. Necessary and sufficient conditions under which 2 × 2 matrices over a local ring are attractive. In this section, we extend these known results on strongly clean matrices to perfect cleanness.

Lemma 3.1. Let R be a ring, and u ∈ U(R). Then the following are equivalent:

(1) a ∈ R is perfectly clean. (2) uau^{−1} ∈ R is perfectly clean.

Proof. (1) ⇒ (2) By hypothesis, there exists an idempotent e ∈ comm^{2}(a) such that a−e ∈ U(R). Hence, uau^{−1} −ueu^{−1} ∈ U(R). For any x ∈ comm(uau^{−1} ), we see that x(uau^{−1} ) = (uau^{−1} )x, and so (u^{−1} xu)a = a(u^{−1} xu). Thus, (u^{−1} xu)e = e(u^{−1} xu). Hence x(ueu^{−1} ) = (ueu^{−1} )x. We conclude that ueu^{−1} ∈ comm^{2}(uau^{−1} ), as required.

(2) ⇒ (1) is symmetric. □

A ring is weakly cobleached provided that for any a ∈ J(R), b ∈ 1 + J(R), l_{a} − r_{b} and l_{b} − r_{a} are both injective. For instance, every commutative local ring, every local ring with nil Jacobson radical.

Theorem 3.1. Let R be a weakly cobleached local ring. Then the following are equivalent:

(1) M_{2}(R) is perfectly clean. (2) M_{2}(R) is strongly clean.

(3) For any A ∈ M_{2}(R), A ∈ GL_{2}(R), or I_{2} − A ∈ GL_{2}(R), or A is similar to a diagonal matrix.

Proof. (1) ⇒ (2) is trivial.

(2) ⇒ (3) is obtained by [13, Theorem 7].

(3) ⇒ (1) For any A ∈ M_{2}(R), A ∈ GL_{2}(R), or I_{2} − A ∈ GL_{2}(R), or A is similar to a diagonal matrix. If A or I_{2} − A ∈ GL_{2}(R), then A is perfectly clean. Assume now that A is similar to a diagonal matrix with A, I_{2} − A ∉ GL_{2}(R). We may assume that A is similar to ( \begin{matrix}λ & 0 \\ 0 & μ \end{matrix} ), where λ ∈ U(R), μ ∈ J(R). If λ ∈ 1+U(R), then ( \begin{matrix}λ & 0 \\ 0 & μ \end{matrix} ) − I_{2} ∈ GL_{2}(R); hence, it is perfectly clean. In view of Lemma 3.1, A is perfectly clean. Thus, we assume that λ ∈ 1 + J(R). By Lemma 3.1, it will suffice to show that ( \begin{matrix}λ & 0 \\ 0 & μ \end{matrix} ) ∈ GL_{2}(R) is perfectly clean. Clearly,

( \begin{matrix}λ & 0 \\ 0 & μ \end{matrix}) = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix}) + ( \begin{matrix}λ & 0 \\ 0 & μ − 1 \end{matrix} ) ,

where ( \begin{matrix}λ & 0 \\ 0 & μ−1 \end{matrix} ) ∈ GL_{2}(R).

We show that the idempotent ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) ∈ comm^{2} (( \begin{matrix}λ & 0 \\ 0 & μ \end{matrix} )). For any ( \begin{matrix}x & s \\ t & y \end{matrix} ) ∈ comm(( \begin{matrix}λ & 0 \\ 0 & μ \end{matrix} )), one has λs = sμ and μt = tλ; hence, s = t = 0. This implies

( \begin{matrix}x & s \\ t & y \end{matrix}) ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix}) = ( \begin{matrix}0 & 0 \\ 0 & y \end{matrix}) = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix}) ( \begin{matrix}x & s \\ t & y \end{matrix}) .

Therefore ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) ∈ comm^{2} (( \begin{matrix}λ & 0 \\ 0 & μ \end{matrix} )), hence the result. □

Corollary 3.1. Let R be a commutative local ring. Then the following are equivalent:

(1) M_{2}(R) is perfectly clean. (2) M_{2}(R) is strongly clean.

(3) For any A ∈ M_{2}(R), A ∈ GL_{2}(R), or I_{2} − A ∈ GL_{2}(R), or A is similar to a diagonal matrix.

Proof. It is a consequence of Theorem 3.1 as every commutative local ring is weakly cobleached. □

Let p be a prime. We use ℤ̂_{p} to denote the ring of all p-adic integers. In view of [6, Theorem 2.4], M_{2}(ℤ̂_{p}) is strongly clean, and therefore M_{2}(ℤ̂_{p}) is perfectly clean, by Corollary 3.1.

Theorem 3.2. Let R and S be local rings. Then the following are equivalent:

(1) ( \begin{matrix}R & V \\ 0 & S \end{matrix} ) is perfectly clean.

(2) For any a ∈ J(R), b ∈ 1 + J(S), v ∈ V, there exists a unique x ∈ V such that ax − xb = v.

Proof. (1) ⇒ (2) Let a ∈ 1 + J(R), b ∈ J(S) and v ∈ V. Set A = ( \begin{matrix}a & −v \\ 0 & b \end{matrix} ). By hypothesis, we can find an idempotent E ∈ comm^{2}(A) such that A − E ∈ ( \begin{matrix}R & V \\ 0 & S \end{matrix} ) is invertible. Clearly, E = ( \begin{matrix}0 & x \\ 0 & 1 \end{matrix} ) for some x ∈ V. Thus, ax − xb = v. Suppose that ay − yb = v for a y ∈ V. Then

A ( \begin{matrix}0 & y \\ 0 & 1 \end{matrix}) = ( \begin{matrix}0 & y \\ 0 & 1 \end{matrix}) A,

and so ( \begin{matrix}0 & y \\ 0 & 1 \end{matrix} ) ∈ comm(A). This implies that

E ( \begin{matrix}0 & y \\ 0 & 1 \end{matrix}) = ( \begin{matrix}0 & y \\ 0 & 1 \end{matrix}) E;

hence, x = y. Therefore there exists a unique x ∈ V such that ax − xb = v, as desired.

(2) ⇒ (1) Let T = ( \begin{matrix}R & V \\ 0 & S \end{matrix} ), and let A = ( \begin{matrix}a & v \\ 0 & b \end{matrix}) ∈ ( \begin{matrix}R & V \\ 0 & S \end{matrix} ).

Case I. a ∈ J(R), b ∈ J(S). Then A − ( \begin{matrix}1_{R} & 0 \\ 0 & 1_{S} \end{matrix} ) ∈ U(T); hence, A is perfectly clean.

Case II. a ∈ U(R), b ∈ U(S). Then A − 0 ∈ U(T); hence, A is perfectly clean. Case III. a ∈ U(R), b ∈ J(S). (i) a ∈ 1+U(R), b ∈ J(S). Then A−( \begin{matrix}1_{R} & 0 \\ 0 & 1_{S} \end{matrix} ) ∈ T

is invertible; hence, A ∈ T is perfectly clean. (ii) a ∈ 1 + J(R), b ∈ J(S). Then we can find a t ∈ V such that at − tb = −v. Let ( \begin{matrix}x & s \\ 0 & y \end{matrix} ) ∈ comm(A). Then

A ( \begin{matrix}x & s \\ 0 & y \end{matrix}) = ( \begin{matrix}x & s \\ 0 & y \end{matrix}) A,

and so ax = xa, by = yb, and as − sb = xv − vy. Hence, we check that

a(xt − ty + s) − (xt − ty + s)b = x(at − tb) − (at − tb)y + (as − sb)
= −xv + vy + (as − sb)
= 0.

By hypothesis, xt − ty = −s, and so we get

( \begin{matrix}0 & t \\ 0 & 1 \end{matrix}) ( \begin{matrix}x & s \\ 0 & y \end{matrix}) = ( \begin{matrix}0 & ty \\ 0 & y \end{matrix}) = ( \begin{matrix}0 & xt + s \\ 0 & y \end{matrix} ) = ( \begin{matrix}x & s \\ 0 & y \end{matrix}) ( \begin{matrix}0 & t \\ 0 & 1 \end{matrix}) .

We infer that

( \begin{matrix}0 & t \\ 0 & 1 \end{matrix}) ^{2} − ( \begin{matrix}0 & t \\ 0 & 1 \end{matrix}) ∈ comm^{2}(A).

Furthermore, A − ( \begin{matrix}0 & t \\ 0 & 1 \end{matrix} ) ∈ U(T). Therefore A is perfectly clean.

Case IV. a ∈ J(R), b ∈ U(S) Then A is perfectly clean, as in the preceding discussion. □

A ring R is uniquely weakly bleached provided that for any a ∈ J(R), b ∈ 1 + J(R), l_{a} − r_{b} and l_{b} − r_{a} are both isomorphisms.

Corollary 3.2. Let R be local. Then the following are equivalent:

(1) T_{2}(R) is perfectly clean. (2) R is uniquely weakly bleached.

Proof. It is clear by Theorem 3.2. □

For instance, if R is a commutative local ring or a local ring with nil Jacobson radical, then T_{2}(R) is perfectly clean.

4. Perfectly J-clean rings

An element a ∈ R is said to be perfectly J-clean provided that there exists an idempotent e ∈ comm^{2}(a) such that a − e ∈ J(R). A ring R is perfectly J-clean if every element in R is perfectly J-clean.

Theorem 4.1. Let R be a ring. Then R is perfectly J-clean if and only if

(1) R is quasipolar. (2) R/J(R) is Boolean.

Proof. Suppose that R is perfectly J-clean. Let a ∈ R is perfectly J-clean. Then there exists an idempotent e ∈ comm^{2}(a) such that w := a−e ∈ J(R). Hence, a − (1 − e) = 2e − 1 + w ∈ U(R). Additionally, (1 − e)a = (1 − e)w ∈ J(R) ⊆ R^{qnil} . This implies that a ∈ R is quasipolar. Furthermore, a − a^{2} = (e + w) − (e + w)^{2} ∈ J(R), and then R/J(R) is Boolean.

Conversely, assume that (1) and (2) hold. Let a ∈ R. Then there exists an idempotent e ∈ comm^{2}(a) such that u := a − e ∈ U(R). Moreover, R/J(R) is Boolean, and so a − a^{2} = (e + u) − (e + u)^{2} = u(1 − 2e − u) ∈ J(R). This shows that 1 − 2e − u ∈ J(R), whence a − (1 − e) = (e + u) − (1 − e) = 2e − 1 + u ∈ J(R). Therefore R is perfectly J-clean. □

Corollary 4.1. Let R be a ring. Then the following are equivalent:

(1) R is perfectly J-clean.

(2) R is perfectly clean, and R/J(R) is Boolean.

(3) R is quasipolar, and R is strongly J-clean.

Proof. (1) ⇒ (2) is obvious by Theorem 4.1, as every quasipolar ring is perfectly clean.

(2) ⇒ (1) For any a ∈ R there exists an idempotent p ∈ comm^{2}(a) such that u := a − p ∈ U(R). As R/J(R) is Boolean, we have ū = ū^{2}; hence, u ∈ 1 + J(R). Furthermore, 2 ∈ J(R). Accordingly, a = p + u = (1 − p) + (2p − 1 + u) with 1 − p ∈ comm^{2}(a) and 2p − 1 + u ∈ J(R), as desired.

(1) ⇒ (3) Suppose R is perfectly J-clean. Then R is strongly J-clean. By the preceding discussion, R is quasipolar.

(3) ⇒ (1) Since R is strongly J-clean, R/J(R) is Boolean. Therefore the proof is complete by the discussion above. □

Example 4.1. Let R = T_{2}(ℤ_{2^{n}} ) (n ∈ ℕ). Then T_{2}(R) is perfectly J-clean.

Proof. As R is finite, it is periodic. This shows that R is strongly π-regular. Hence, T_{2}(R) is quasipolar, by [9, Theorem 2.6]. As J(ℤ_{2^{n}} ) = 2ℤ_{2^{n}} , we see that R/J(R) ≅ ℤ_{2} is Boolean. Hence, T_{2}(R)/J(T_{2}(R)) is Boolean. Therefore the result follows by Theorem 4.1. □

Recall that a ring R is uniquely strongly clean provided that for any a ∈ R there exists a unique idempotent e ∈ comm(a) such that a − e ∈ U(R).

Proposition 4.1. Let R be a ring. Then R is perfectly J-clean if and only if (1) R is perfectly clean, (2) R is uniquely strongly clean.

Proof. Suppose R is perfectly J-clean. Then R is perfectly clean. Hence, R is strongly clean. Let a ∈ R. Write a = e + u = f + v with e = e^{2} ∈ comm^{2}(a), f = f^{2} ∈ R, u ∈ J(R), v ∈ U(R), ea = ae and fa = af. Then f ∈ comm(a), and so ef = fe. Thus, e − f = v − u ∈ U(R) and (e − f)(e + f − 1) = 0. This implies that f = 1 − e, and therefore R is uniquely strongly clean.

Conversely, assume that (1) and (2) hold. Then R/J(R) is Boolean. Therefore we complete the proof by Corollary 4.1. □

Corollary 4.2. A ring R is uniquely clean if and only if R is abelian perfectly J-clean.

Proof. As every uniquely clean ring is abelian (cf. [4, Corollary 16.4.16]), it is clear by Proposition 4.1. □

Theorem 4.2. Let R be a ring. Then the following are equivalent:

(1) R is perfectly J-clean.

(2) For any a ∈ R, there exists a unique idempotent e ∈ comm^{2}(a) such that a − e ∈ J(R).

Proof. (1) ⇒ (2) For any a ∈ R, there exists an idempotent e ∈ comm^{2}(a) such that a−e ∈ J(R). Assume that a−f ∈ J(R) for an idempotent f ∈ comm^{2}(a). Clearly, e ∈ comm^{2}(a) ⊆ comm(a). As f ∈ comm^{2}(a), we see that ef = fe. Thus, (e−f)^{3} = e−f, and so (e−f)(1−(e−f)^{2}) = 0. But e−f = (a−f)−(a−e) ∈ J(R), as a − f, a − e ∈ J(R). Hence, e = f, as desired.

(2) ⇒ (1) is trivial. □

Recall that a ring R is strongly J-clean provided that for any a ∈ R there exists an idempotent e ∈ comm(a) such that a − e ∈ J(R) (cf. [3, 4]).

Corollary 4.3. A ring R is perfectly J-clean if and only if

(1) R is quasipolar, (2) R is strongly J-clean.

Proof. Suppose R is perfectly J-clean. Then R is strongly J-clean. For any a ∈ R, there exists an idempotent p ∈ comm^{2}(a) such that w := a − p ∈ J(R). Hence, a = (1 − p) + (2p − 1 + w) with 1 − p ∈ comm^{2}(a) and 2p − 1 + w ∈ U(R). Furthermore, (1 − p)a = (1 − p)w ∈ J(R) ⊆ R^{qnil} . Therefore, R is quasipolar.

Conversely, assume that (1) and (2) hold. Since R is quasipolar, it is perfectly clean. By virtue of [4, Proposition 16.4.15], R/J(R) is Boolean. Therefore the proof is complete by Corollary 4.1. □

Following Cui and Chen [8], a ring R is called J-quasipolar provided that for any element a ∈ R there exists an e ∈ comm^{2}(a) such that a + e ∈ J(R). We further show that the two concepts coincide. But this is not the case for a single element. That is,

Proposition 4.2. A ring R is perfectly J-clean if and only if for any element a ∈ R there exists an e ∈ comm^{2}(a) such that a + e ∈ J(R).

Proof. Let R be perfectly J-clean. Then R/J(R) is Boolean, by Theorem 4.1. Hence, ¯2^{2} = ¯2, i.e., 2 ∈ J(R). For any a ∈ R, there exists an idempotent e ∈ comm^{2}(a) such that a − e ∈ J(R). This implies that a + e = (a − e) + 2e ∈ J(R). The converse is similar by [8, Corollary 2.3]. □

Example 4.2. Let R = ℤ_{3}. Note that J(R) = 0. Since ¯1 − ¯1 = ¯0 ∈ J(R), ¯1 is

perfectly J-clean, but we can not find an idempotent e ∈ R such that ¯1+ e ∈ J(R), because ¯1 + ¯0 ∉ J(R) and ¯1 + ¯1 = ¯2 ∉ J(R).

Further, though ¯2 + ¯1 = ¯0 ∈ J(R), we can not find an idempotent e ∈ R such that ¯2 − e ∈ J(R), because ¯2 − ¯0 = ¯2 ∉ J(R) and ¯2 − ¯1 = ¯1 ∉ J(R).

Lemma 4.1. Let R be a ring. Then a ∈ R is perfectly J-clean if and only if (1) a ∈ R is quasipolar, (2) a − a^{2} ∈ J(R).

Proof. Suppose that a ∈ R is perfectly J-clean. Then there exists an idempotent e ∈ comm^{2}(a) such that w := a − e ∈ J(R). Hence, a − (1 − e) = 2e − 1 + w ∈ U(R). Additionally, (1 − e)a = (1 − e)w ∈ J(R) ⊆ R^{qnil} . This implies that a ∈ R is quasipolar. Furthermore, (e + w) − (e + w)^{2} = −(2e − 1 + w)w ∈ J(R).

Conversely, assume that (1) and (2) hold. Then there exists an idempotent e ∈ comm^{2}(−a) such that (−a) + e ∈ U(R). Set u := a − e. Then a − a^{2} = (e + u) − (e + u)^{2} = u(1 − 2e − u) ∈ J(R); hence, 1 − 2e − u ∈ J(R). This shows that a−(1−e) = (e+u)−(1−e) = 2e−1+u ∈ J(R). Therefore a ∈ R is perfectly J-clean. □

Theorem 4.3. Let R be a commutative ring, and let A ∈ T_{n}(R). If 2 ∈ J(R), then the following are equivalent:

(1) A ∈ T_{n}(R) is perfectly J-clean. (2) Each A_{ii} ∈ T_{n}(R) is perfectly J-clean.

Proof. (1) ⇒ (2) is obvious.

(2) ⇒ (1) Clearly, the result holds for n = 1. Suppose that the result holds

for n − 1 (n ⩾ 2). Let A = ( \begin{matrix}a_{11} & α \\ 0 & A_{1} \end{matrix} ) ∈ T_{n}(R) where a_{11} ∈ R, α ∈ M_{1×(n−1)} (R)

and A_{1} ∈ T_{n−1} (R). Then we have an idempotent e_{11} ∈ R such that w_{11} := a_{11} − e_{11} ∈ J(R). By hypothesis, we have an idempotent E_{1} ∈ T_{n−1} (R) such that W_{1} := A_{1} − E_{1} ∈ J(T_{n−1} (R)) and E_{1} ∈ comm^{2}(A_{1}). As 2 ∈ J(R),

W_{1} + (1 − 2e_{11} − w_{11})I_{n−1} ∈ I_{n−1} + J(T_{n−1} (R)) ⊆ U(T_{n−1} (R)).

Let E = ( \begin{matrix}e_{11} & β \\ 0 & E_{1} \end{matrix} ), where β = α(E_{1} − e_{11}I_{n−1} )(W_{1} + (1 − 2e_{11} − w_{11})I_{n−1} )^{−1} . Then

A − E ∈ J(T_{n}(R)). As

e_{11}β + βE_{1} = β(E_{1} + e_{11}I_{n−1} )
= α(E_{1} − e_{11}I_{n−1} )(E_{1} + e_{11}I_{n−1} )(W_{1} + (1 − 2e_{11} − w_{11})I_{n−1} )^{−1} = β,

we see that E = E^{2}.

For any X = ( \begin{matrix}x_{11} & γ \\ 0 & X_{1} \end{matrix} ) ∈ comm(A), we have x_{11}α + γA_{1} = a_{11}γ + αX_{1}; hence,

α(X_{1} − x_{11}I_{n−1} ) = γ(A_{1} − a_{11}I_{n−1} ).

As E_{1} ∈ comm^{2}(A_{1}), we get

γ(A_{1} − a_{11} I_{n−1} )(E_{1} − e_{11}I_{n−1} )
= α(X_{1} − x_{11}I_{n−1} )(E_{1} − e_{11}I_{n−1} )
= α(E_{1} − e_{11}I_{n−1} )(X_{1} − x_{11}I_{n−1} )
= β(W_{1} + (1 − 2e_{11} − w_{11})I_{n−1} )(X_{1} − x_{11}I_{n−1} )
= β(X_{1} − x_{11}I_{n−1} )(W_{1} + (1 − 2e_{11} − w_{11})I_{n−1} ).

Furthermore,

γ(A_{1} − a_{11} I_{n−1} )(E_{1} − e_{11}I_{n−1} )
= γ(E_{1} − e_{11}I_{n−1} )(E_{1} + W_{1} − (e_{11} + w_{11})I_{n−1} )
= γ(E_{1} − e_{11}I_{n−1} )(E_{1} + e_{11}I_{n−1} + (W_{1} − 2e_{11} − w_{11})I_{n−1} )
= γ(E_{1} − e_{11}I_{n−1} + (E_{1} − e_{11}I_{n−1} )(W_{1} − 2e_{11} − w_{11})I_{n−1} )
= γ(E_{1} − e_{11}I_{n−1} )(W_{1} + (1 − 2e_{11} − w_{11})I_{n−1} ).

It follows from W_{1} + (1 − 2e_{11} − w_{11})I_{n−1} ∈ U(T_{n−1} (R)) that γ(E_{1} − e_{11}I_{n−1} ) = β(X_{1} − x_{11}I_{n−1} ). Hence, e_{11}γ + βX_{1} = x_{11}β + γE_{1}, and so EX = XE. This implies that E ∈ comm^{2}(A). By induction, A ∈ T_{n}(R) is perfectly J-clean for all n ∈ ℕ. □

Corollary 4.4. Let R be a commutative ring. Then the following are equivalent:

(1) R is strongly J-clean.

(2) T_{n}(R) is perfectly J-clean for all n ∈ ℕ.

(3) T_{n}(R) is perfectly J-clean for some n ∈ ℕ.

Proof. (1) ⇒ (2) As R is strongly J-clean, R/J(R) is Boolean. Hence, 2 ∈ J(R). For any n ∈ ℕ, T_{n}(R) is perfectly J-clean by Theorem 4.3.

(2) ⇒ (3) ⇒ (1) These are clear by Theorem 4.3. □

Let R be Boolean. As a consequence of Corollary 4.4, T_{n}(R) is perfectly J-clean for all n ∈ ℕ.

Lemma 4.2. Let R be a ring, and u ∈ U(R). Then the following are equivalent:

(1) a ∈ R is perfectly J-clean. (2) uau^{−1} ∈ R is perfectly J-clean.

Proof. (1) ⇒ (2) As in the proof of Lemma 3.1, uau^{−1} ∈ R is quasipolar. Furthermore, uau^{−1} − (uau^{−1} )^{2} = u(a − a^{2})u^{−1} ∈ J(R). As in the proof of Theorem 4.1, uau^{−1} ∈ R is perfectly J-clean.

(2) ⇒ (1) is symmetric. □

We end this paper by showing that strong J-cleanness of 2 × 2 matrix ring over a commutative local ring can be enhanced to perfect J-cleanness.

Theorem 4.4. Let R be a commutative local ring, and let A ∈ M_{2}(R). Then the following are equivalent:

(1) A is perfectly J-clean. (2) A is strongly J-clean.

(3) A ∈ J(M_{2}(R)), or I_{2} − A ∈ J(M_{2}(R)), or the equation x^{2} − tr(A)x + det(A) = 0 has a root in J(R) and a root in 1 + J(R).

Proof. (1) ⇒ (2) is trivial.

(2) ⇒ (3) is proved by [4, Theorem 16.4.31].

(3) ⇒ (1) If A ∈ J(M_{2}(R)) or I_{2} − A ∈ J(M_{2}(R)), then A is perfectly J-clean. Otherwise, it follows from [4, Theorem 16.4.31 and Proposition 16.4.26] that there exists a U ∈ GL_{2}(R) such that

UAU^{−1} = ( \begin{matrix}α \\ & β \end{matrix}) = ( \begin{matrix}0 \\ & 1 \end{matrix}) + ( \begin{matrix}α \\ & β − 1 \end{matrix} ) ,

where α ∈ J(R), β ∈ 1 + J(R). For any X ∈ comm(UAU^{−1} ), we have X( \begin{matrix}α \\ & β \end{matrix} ) = ( \begin{matrix}α \\ & β \end{matrix} )X; hence, βX_{12} = αX_{12}. This implies that X_{12} = 0. Likewise, X_{21} = 0. Thus,

X ( \begin{matrix}0 \\ & 1 \end{matrix}) = ( \begin{matrix}0 \\ & 1 \end{matrix}) X,

and so ( \begin{matrix}0 \\ & 1 \end{matrix} ) ∈ comm^{2}(UAU^{−1} ). As a result, UAU^{−1} is perfectly J-clean, and then so is A by Lemma 4.2. □

Corollary 4.5. Let R be a commutative local ring. Then the following are equivalent:

(1) M_{2}(R) is perfectly clean.

(2) For any A ∈ M_{2}(R), A ∈ GL_{2}(R), or I_{2} − A ∈ GL_{2}(R), or A ∈ M_{2}(R) is perfectly J-clean.

Proof. (1) ⇒ (2) is proved by Theorem 3.1, [4, Corollary 16.4.33] and Theorem 4.4.

(2) ⇒ (1) is obvious. □

Acknowledgement. The authors are grateful to the referee for his/her helpful suggestions. H. Chen is thankful for the support by the Natural Science Foundation of Zhejiang Province, China (No. LY17A010018).

References

1. G. Borooah, A. J. Diesl, T. J. Dorsey, Strongly clean triangular matrix rings over local rings, J. Algebra 312 (2007), 773–797.

2. , Strongly clean matrix rings over commutative local rings, J. Pure Appl. Algebra 212 (2008), 281–296.

3. H. Chen, On strongly J-clean rings, Commun. Algebra 38 (2010), 3790–3804.

4. , Rings Related Stable Range Conditions, Series in Algebra 11, World Scientific, Hackensack, New Jersey, 2011.

5. J. Chen, X. Yang, Y. Zhou, When is the 2 × 2 matrix ring over a commutative local ring strongly clean?, J. Algebra 301 (2006), 280–293.

6. , On strongly clean matrix and triangular matrix rings, Commun. Algebra 34 (2006), 3659–3674.

7. J. Cui, J. Chen, Quasipolar triangular matrix rings over local rings, Commun. Algebra 40 (2012), 784–794.

8. , A class of quasipolar rings, Commun. Algebra 40 (2012), 4471–4482.

9. , Characterizations of quasipolar rings, Commun. Algebra 41 (2013), 3207–3217.

10. A. J. Diesl, Nil clean rings, J. Algebra 383 (2013), 197–211.

11. M. P. Drazin, A class of outer generalized inverses, Linear Algebra Appl. 436 (2012), 1909–1923.

12. Y. Li, Strongly clean matrix rings over local rings, J. Algebra 312 (2007), 397–404.

13. X. Yang, Y. Zhou, Strong cleanness of the 2 × 2 matrix ring over a general local ring, J. Algebra 320 (2008), 2280–2290.

Department of Mathematics (Received 26 03 2015) Hangzhou Normal University (Revised 18 07 2016) Hangzhou, China

huanyinchen@aliyun.com

Department of Mathematics Ankara University Ankara, Turkey

halici@ankara.edu.tr

Department of Mathematics Ahi Evran University Kirsehir, Turkey

handankose@gmail.com