1　Introduction

Throughout this paper, R denotes a ring with an identity. J(R) and U(R) will denote, respectively, the Jacobson radical and the group of units in R. And J#(R)={x∈R|xn∈J(R)}. For an element a∈R, we define the commutant and double commutant of a by comm(a)={x∈R|ax=xa}, comm2(a)={x∈R|xy=yx,∀y∈comm(a)} , respectively.

From [1] Cui Jian and Chen Jianlong called a∈R is J-quasipolar if there exists p2=p∈comm2(a) such that a+p∈J(R). The idempotent p satisfying the above conditions is called J-spectral idempotent of a. A ring R is said to be J-quasipolar if each element of R is J-quasipolar. For convenience, we list several results which will be used in the sequel.

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We called idempotents lift (*) modulo I if a2-a∈I, then there exists p2=p∈comm2(a) such that e-a∈I .

Theorem 1[1,Corollary2.3] If R is J-quasipolar, then 2∈J(R).

Theorem 2[1,Theorem2.9] Let R be a ring. Then the following conditions are equivalent.

(1) R is a J-quasipolar.

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(2) R/J(R) is Boolean and idempotents lift (*) modulo J(R).

(3) R/J(R) is Boolean and R is quasipolar .

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Theorem 3[2,Theorem2.7] A ring R is strongly nil-clean if R/J(R) is boolean and J(R) is nil.

2　Structure theorem

In this section, we prove a new characteration of the properties of J-quasipolar rings.

Theorem 4　Let R be a ring. Then the following conditions are equivalent.

Corollary 2　Let R be a ring. Then the following conditions are equivalent:

(2) For any a∈R , there exists p2=p∈comm2(a) such that a+p∈J#(R) .

(3) For any a∈R , there exists p2=p∈comm2(a) such that a-p∈J#(R) .

Proof　(1)⟹(2) is clear .

(2)⟹(1). Let a∈R, then consider -a∈R , we have -a+p∈J#(R), let -a+p=w, where wn∈J(R), set it is clear that since wn∈J(R), then is nilpotent of since p∈comm2(a), then Thus is strongly nil clean. By Theorem 3, is Boolean. Since ≅ then we have is Boolean, and it is obvious that idempotents lift (*) modulo J(R). By Theorem 2, R is J-quasipolar.

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(1)⟺(3) Applying the preceding argument to -a, we easily ontain the result.

Corollary 1　Let R be a ring. Then R is a J-quasipolar ring if and only if

(1) R/J(R) is Boolean;

(2) If a-a2∈J#(R), then there exists p2=p∈comm2(a) such that a-p∈J#(R) .

Proof　 ⟸ is clear.

⟹ for each a∈R, since R/J(R) is Boolean, then we have that is a-a2∈J(R), thus a-a2∈J#(R), then there exists p2=p∈comm2(a) such that a-p∈J#(R), by Theorem 4, R is J-quasipolar.

Theorem 5　R is a J-quasipolar ring if and only if

(1) 2∈J(R);

(2) For each a∈R,there exists p2=p∈comm2(a) such that a-p∈J(R) or a+p∈J(R).

Proof　⟸ is clear.

(2) For any a∈R, there exist p2=p∈comm2(a) such that ap∈J#(R), (1-p)(1-a)∈J#(R).

Example 1　In Theorem 5, the condition 2∈J(R) is essential. Let R=Z3, then for each a∈R,there exists e2=e∈comm2(a) such that a-e∈J(R) or a+e∈J(R). But 2∉J(R), Thus R is not J-quasipolar.

(1) R is a J-quasipolar ring.

(1) R is a J-quasipolar ring.

(2) 2∈J(R) and for any a∈R, there exists p2=p∈comm2(a) such that a2-p∈J(R).

Proof　By Theorem 5, we know that (1)⟹(2) is clear. Then consider (2)⟹(1), since a2-e∈J(R), we have (a+e)(a-e)∈J(R),that is (a+e)(a+e-2e)∈J(R),(a+e)2-2e(a+e)∈J(R). Since 2∈J(R), then (a+e)2∈J(R), that is a+e∈J#(R). By Theorem 4, R is a J-quasipolar ring.

(2) For any a∈R , there exists p2=p∈comm2(a) such that a+p,a-p∈J(R).

(1) R is a J-quasipolar ring.

(2)⟹(3) This is trivial.

⟹ Suppose for each a∈R,there exists e2=e∈comm2(a) such that a-e∈J(R) or a+e∈J(R) and 2∈J(R). That is (a-e)(a+e)∈J(R),(a+e-2e)(a+e)∈J(R),(a+e)2-2e(a+e)∈J(R). Since 2∈J(R), then (a+e)2∈J(R), that is a+e∈J#(R). From Theorem 4 we know that R is J-quasipolar.

Proof　(1)⟹(2) is clear.

(2)⟹(1), We have ap∈J#(R), (1-p)(1-a)∈J#(R). Consider (ap-(1-p)(1-a))k. Because there exists k,l∈N such that (ap)k∈J(R) and ((1-p)(1-a))l∈J(R). Choose m=max(k,l). Then (ap-(1-p)(1-a))m∈J(R), that is (a-(1-p))m∈J(R). Set f=1-p, since p2=p∈comm2(a), we check that f2=f∈comm2(a) and a-f∈J#(R), From Theorem 4, R is a J-quasipolar ring.

Recall that a ring R is strongly J#-clean if for any a∈R, there exists an idempotent p∈R such that a-p∈J#(R) and ap=pa.

Theorem 7　A ring R is J-quasipolar ring if and only if

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(1) R is a quasipolar ring;

(2) R is a strongly J#-clean ring.

(1) R is a nil-quasipolar ring.

⟹ Let a∈R. By (2), there exists e=e2∈comm(a) such that (a-e)n∈J(R). By (1), we can find f=f2∈comm2(a) such that a-f∈U(R). This implies that ef=fe, and so e-f=(a-f)-(a-e)=(a-f)(1-(a-f)-1(a-e))∈U(R). Since (e-f)(e+f)=e-f, we see that e+f=1, and so e=1-f∈comm2(a). Therefore R is J-quasipolar by Theorem 4.

3　Extensions

The main purpose of this section is to investigate the necessary and sufficient conditions for T2(R) over a commutative local ring R to be J-quasipolar.

Theorem 8　Let R be a commutative local ring. Then the following conditions are equivalent.

(1) T2(R) is a J-quasipolar ring.

(2) For any A∈T2(R) , there exist E2=E∈comm2(A) and A+E∈J(T2(R)) or A-E∈J(T2(R)).

(3) R is a J-quasipolar ring.

Proof　(1)⟹(2) is clear.

(2)⟹(3) ∀a∈R,write then A∈T2(R), there exists E2=E∈comm2(A) and A+E∈J(T2(R)) or A-E∈J(T2(R)), write where e2=e∈comma2(a),f2=f∈comma2(a),x∈R.

Case I　if A+E∈J(T2(R)), then we have thus a+e,-a+f∈J(R), thus there exists f2=f∈comma2(a) such that a-f∈J(R),R is J-quasipolar.

Case II　if A-E∈J(T2(R)), then thus a-e,-a-f∈J(R), thus there exists e2=e∈comma2(a) such that a-e∈J(R),R is J-quasipolar.

(3)⟹(1) Since R is local and J-quasipolar, by [1,Theorem 2.11] we know R/J(R)≅Z2. And because R is commutative and local, from [2] we know that R is uniquely bleached. Then because R is uniquely bleached and R/J(R)≅Z2, by [1,Theorem 3.3] we have T2(R) is J-quasipolar.

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Theorem 9　Let R be a ring. Then the following conditions are equivalent.

(1) R is a J-quasipolar ring.

Theorem 6　Let R be a ring. Then the following conditions are equivalent:

(3) For any a∈R , there exists p2=p∈comm2(a) such that a+p,a-p∈J#(R).

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Proof　(1)⟹(2) Clearly, 2∈J(R). let a∈R. Then there exists p2=p∈comm2(a) such that a+p∈J(R). Thus, a-p=(a+p)-2p∈J(R), as desired.

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(3)⟹(1) This is obvious by Theorem 4.

Let R be an arbitrary ring. An element a∈R is nil-quasipolar if there exists p2=p∈comm2(a) such that a+p∈N(R). A ring R is called nil-quasipolar in case each of its elements is nil-quasipolar [3]. Further, we derive

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Corollary 3　Let R be a ring. Then the following conditions are equivalent.

Proof　⟸ is clear.

(2) For any a∈R , there exists p2=p∈comm2(a) such that a+p,a-p∈N(R).

(3) J(R) is nil and R is a J-quasipolar ring.

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Proof　(1)⟹(3) Clearly, J(R) is nil. Let a∈R. Then there exists p2=p∈comm2(a) such that a+p∈N(R). Hence, there are some n∈ such that (a+p)n=0∈J(R). In light of Theorem 4, R is a J-quasipolar.

(2)⟹(3) Let a∈R. In light of Theorem 9, there exists p2=p∈comm2(a) such that a+p,a-p∈J(R). As J(R)⊆N(R), we see that a+p,a-p∈N(R), as required.

(3)⟹(1) This is clear by the definition.

Corollary 4　Let R be a commutative local ring. Then the following conditions are equivalent.

(1) T2(R) is a nil-quasipolar ring.

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(2) For any A∈T2(R) , there exist E2=E∈comm2(A) and A+E∈N(T2(R)) or A-E∈N(T2(R)).

(3) R is a nil-quasipolar ring.

Proof　We easily obtain the result, by Corollary 3 and Theorem 8.

Recall that a ring R is strongly nil-clean if every element in R is the sum of an idempotent and a nilpotent that commute [4]. Finally, we note that strongly nil-clean ring and nil-quasipolar ring coincide with each other. That is, we have

Proposition 1　A ring R is strongly nil-clean if and only if R is nil-quasipolar.

Proof　⟹ Let a∈R. Then we can find an idempotent e∈comm(a) such that a-e∈N(R). Hence, a-a2∈N(R). By the lifting property of idempotent, we can find some e∈[a] such that a-e∈N(R). Write e=f(a) for some f(t)∈[t]. If c∈comm(a), then ec=f(a)c=cf(a)=ce. This implies that e∈comm2(a). Therefore R is nil-quasipolar, as asserted.

⟸ This is obvious.

References:

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[2] BOROOAH G, DIESL A J, DORSEY T J. Strongly clean triangular matrix rings over local rings[J]. J Algebra,2007, 312(2): 773-797.

[3] GURGUN O, HALICIOGLU S, HARMANCI A. Nil-quasipolar rings[J]. Boletin de la Sociedad Matematica Mexicana,2014, 20(1): 29-38.

[4] KOSAN T, WANG Z, ZHOU Y. Nil-clean and strongly nil-clean rings[J]. J Algebra Appl,2016, 220(2): 633-646.