Rend. Circ. Mat. PalermoDOI 10.1007/s12215-014-0153-y

Additive mappings satisfying algebraic conditions in rings

Abu Zaid Ansari · Faiza Shujat

Received: 5 March 2014 / Accepted: 29 March 2014© Springer-Verlag Italia 2014

Abstract Let R be any (n + 1)!-torsion free ring and F, D : R → R be additive mappingssatisfying F(xn+1) = (α(x))n F(x) + ∑n

i=1(α(x))n−i (β(x))i D(x) for all x ∈ R, where nis a fixed integer and α, β are automorphisms of R. Then, D is Jordan left (α, β)-derivationand F is generalized Jordan left (α, β)-derivation on R and if additive mappings F and Dsatisfying F(xn+1) = F(x)(α(x))n + ∑n

i=1(β(x))i D(x)(α(x))n−i for all x ∈ R. Then, Dis Jordan (α, β)-derivation and F is generalized Jordan (α, β)-derivation on R. At last someimmediate consequences of the above theorems have been given.

Keywords Generalized Jordan (left) (α, β)-derivation · Jordan (left) (α, β)-derivation ·Automorphisms

Mathematics Subject Classification (2000) 16W25 · 16N60 · 16U80

1 Introduction

Throughout the present paper R will denote an associative ring with identity. A ring R isn-torsion free, where n > 1 is an integer, in case nx = 0, x ∈ R implies x = 0. Recall thatR is prime if a Rb = {0} implies a = 0 or b = 0, and is semiprime if a Ra = {0} impliesa = 0. An additive mapping d : R → R is called a derivation if d(xy) = d(x)y + xd(y)

for all x, y ∈ R and is called a Jordan derivation in case d(x2) = d(x)x + xd(x) is fulfilledfor all x ∈ R. Every derivation is a Jordan derivation, but the converse need not be true in

A. Z. Ansari (B)Department of Mathematics, Faculty of Science, Islamic University in Madinah,Madinah, Kingdom of Saudi Arabiae-mail: ansari.abuzaid@gmail.com

F. ShujatDepartment of Applied Mathematics, Faculty of Engineering, Aligarh Muslim University,Aligarh, Indiae-mail: faiza.shujat@gmail.com

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A. Z. Ansari, F. Shujat

general. A classical result due to Herstein [9, Theorem 3.3], asserts that a Jordan derivationon a prime ring of characteristic different from two is a derivation. A brief proof of Herstein’stheorem can be found in [7]. Cusack [7] has extended Herstein’s theorem for 2-torsion freesemiprime ring. An additive mapping F : R → R is said to be a generalized derivation ifthere exists a derivation d : R → R such that F(xy) = F(x)y + xd(y) for all x, y ∈ R.An additive mapping D : R → R is said to be a left derivation (resp. Jordan left derivation)if D(xy) = x D(y) + y D(x) (resp. D(x2) = 2x D(x)) holds for all x, y ∈ R. Clearly,every left derivation on a ring R is a Jordan left derivation but the converse need not betrue in general (see Example 1.1 in [13]). In Ashraf and Rehman [4] together with Shakirproved that a Jordan left derivation on a 2-torsion free prime ring is a left derivation. Duringthe last decade, there has been on going interest concerning the relationship between leftderivation and Jordan left derivation on prime and semiprime rings (cf, [1–4,6,8,10,11,13]and reference therein).

Following [12], an additive mapping T : R → R is called left (resp. right) centralizerof R if T (xy) = T (x)y (resp. T (xy) = xT (y)) for all x, y ∈ R. An additive mappingT : R → R is called Jordan left (resp. Jordan right) centralizer of R if T (x2) = T (x)x(resp. T (x2) = xT (x)) for all x ∈ R. Obviously, every left (resp. right) centralizer isa Jordan left (resp. right) centralizer on R. The converse need not be true in general. InZalar [12], proved that every Jordan left centralizer on a 2-torsion free semiprime ring is aleft centralizer. Following [2], an additive mapping F : R → R is called generalized leftderivation (reps. generalized Jordan left derivation) if there exists a Jordan left deviationD : R → R such that F(xy) = x F(y) + y D(x) ( resp. F(x2) = x F(x) + x D(x)) forall x ∈ R. It is easy to see that F : R → R is a generalized left derivation if and only ifF is of the form F = D + T , where D is a left derivation and T right centralizer of R.The concept of generalized left derivations cover the concept of left derivation. However, ageneralized left derivation with D = 0 includes the concept of right centralizer. Since thesum of two generalized left derivations is a generalized left derivation. For a fixed a ∈ R,every map of the form F(x) = xa + D(x), where D is a left derivation of R is a generalizedleft derivation. Notice that for any generalized left derivation F , the mapping G : R → Rsuch that G(x) = F(x) + xa or G(x) = F(x) − xa, where a is a fixed element of R, is alsoa generalized left derivation on R. It is obvious to see that every generalized left derivationon a 2-torsion free ring R is a generalized Jordan left derivation but the converse need not betrue in general (see Example 1.1 of [2]). It is shown in [2] that if R is 2-torsion free primering, then every generalized Jordan left derivation is a generalized left derivation. Recently,Shakir [1] extended this result for semiprime ring.

Suppose α and β are two automorphisms on R. An additive mapping D : R → R is saidto be a left (α, β)-derivation (resp. Jordan left (α, β)-derivation) if D(xy) = α(x)D(y) +β(y)D(x) (resp. D(x2) = α(x)D(x) + β(x)D(x)) holds for all x, y ∈ R. An additivemapping F : R → R is called a generalized left (α, β)-derivation (reps. generalized Jordanleft (α, β)-derivation) if there exists a Jordan left (α, β)-deviation D : R → R such thatF(xy) = α(x)F(y) + β(y)D(x) ( resp. F(x2) = α(x)F(x) + β(x)D(x)) for all x, y ∈ R.An additive mapping D : R → R is said to be an (α, β)-derivation (resp. Jordan (α, β)-derivation) if D(xy) = D(x)α(y) + β(x)D(y) (resp. D(x2) = D(x)α(x) + β(x)D(x))holds for all x, y ∈ R. An additive mapping F : R → R is called generalized (α, β)-derivation (reps. generalized Jordan (α, β)-derivation) if there exists an (α, β)-deviation(Jordan (α, β)-deviation) D : R → R such that F(xy) = F(x)α(y) + β(x)D(y) ( resp.F(x2) = F(x)α(x) + β(x)D(x)) for all x, y ∈ R.

Very recently in 2013 Ashraf et al. [5], considered the additive mappings F, D : R → Rsatisfying the condition F(xn+1) = xn F(x) + nxn D(x) for all x ∈ R, and showed that

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Algebraic conditions in rings

if R is an (n + 1)!-torsion free ring with identity, then D is Jordan left derivation and Fis generalized Jordan left derivation on R. Thus in view of [5, Theorem 1], it is naturalto ask whether the additive mapping satisfying the condition F(xn+1) = (α(x))n F(x) +∑n

i=1(α(x))n−i (β(x))i D(x) forces that F is generalized Jordan left (α, β)-derivation andD is Jordan left (α, β)-derivation on R, where α, β are automorphisms on R. The answer isin affirmative sense. More precisely we prove the following theorem:

Theorem 1.1 Let n ≥ 1 be any fixed integer and R be any (n + 1)!-torsion free ring. Sup-pose that F, D : R → R are additive mappings satisfying F(xn+1) = (α(x))n F(x) +∑n

i=1(α(x))n−i (β(x))i D(x) for all x ∈ R, where n is fixed integer and α, β are endo-morphisms of R. Then, D is Jordan left (α, β)-derivation and F is generalized Jordan left(α, β)-derivation on R.

Proof Given that

F(xn+1) = (α(x))n F(x) +n∑

i=1

(α(x))n−i (β(x))i D(x) for all x ∈ R. (1.1)

Replacing x by e in the above expression, we get nD(e) = 0. Since R is n-torsion free andα(e) = β(e) = e, we obtain D(e) = 0. Again, replacing x by x + ke in (1.1), where k is anypositive integer, we get

F((x + ke)n+1) = (α(x + ke))n F(x + ke)

+n∑

i=1

(α(x + ke))n−i (β(x + ke))i D(x + ke)

= (α(x) + ke)n(F(x) + k F(e))

+n∑

i=1

(α(x) + ke)n−i (β(x) + ke))i D(x) for all x ∈ R.

F[xn+1 +

( n + 11

)xnk +

( n + 12

)xn−1k2 + · · · + kn+1e

]

=[(α(x))n +

( n1

)(α(x))n−1k +

( n2

)(α(x))n−2k2 + · · · + kne

](F(x) + k F(e))

+n∑

i=1

[(α(x))n−i + · · · +

( n − in − i − 2

)(α(x))2kn−i−2

+( n − i

n − i − 1

)α(x)kn−i−1 + kn−i e

]·

[(β(x))i + · · · +

( ii − 2

)(β(x))2ki−2

+( i

i − 1

)(β(x))ki−1 + ki e

]D(x) for all x ∈ R.

F(xn+1) + F[( n + 1

1

)xnk +

( n + 12

)xn−1k2 + · · · + kn+1e

]

= (α(x))n(F(x) + ke) +[( n

1

)(α(x))n−1k +

( n2

)(α(x))n−2k2 + · · ·

+( n

n − 2

)(α(x))2kn−2 +

( nn − 1

)α(x)kn−1 + kne

](F(x) + k F(e))

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A. Z. Ansari, F. Shujat

+n∑

i=1

[(α(x))n−i +

( n − i1

)(α(x))n−i−1k + · · · +

( n − in − i − 2

)(α(x))2kn−i−2

+( n − i

n − i − 1

)α(x)kn−i−1 + kn−i e

].(β(x))i D(x)

+n∑

i=1

[(α(x))n−i +

( n − i1

)(α(x))n−i−1k + . . . +

( n − in − i − 2

)(α(x))2kn−i−2

+( n − i

n − i − 1

)α(x)kn−i−1 + kn−i e

][( i1

)(β(x))i−1k +

( i2

)(β(x))i−2k2

+ · · · +( i

i − 2

)(β(x))2ki−2 +

( ii − 1

)(β(x))ki−1 + ki e

]D(x)

F[( n + 1

1

)xnk +

( n + 12

)xn−1k2 + · · · + kn+1e

]

= (α(x))nke +[( n

1

)(α(x))n−1k +

( n2

)(α(x))n−2k2 + · · · +

( nn − 2

)(α(x))2kn−2

+( n

n − 1

)α(x)kn−1 + kne

](F(x) + k F(e)) +

n∑

i=1

[( n − i1

)(α(x))n−i−1k + · · ·

+( n − i

n − i − 2

)(α(x))2kn−i−2 +

( n − in − i − 1

)α(x)kn−i−1 + kn−i e

].(β(x))i D(x)

+n∑

i=1

[(α(x))n−i +

( n − i1

)(α(x))n−i−1k + . . . +

( n − in − i − 2

)(α(x))2kn−i−2

+( n − i

n − i − 1

)α(x)kn−i−1 + kn−i e

][( i1

)(β(x))i−1k +

( i2

)(β(x))i−2k2

+ · · · +( i

i − 2

)(β(x))2ki−2 +

( ii − 1

)(β(x))ki−1 + ki e

]D(x).

This can be written as

k f1(α(x), β(x), e) + k2 f2(α(x), β(x), e) + · · · + kn fn(α(x), β(x), e) = 0

for all x ∈ R, where fi (α(x), β(x), e) are the coefficients of ki ′s, for all i = 1, 2, . . . , n. Now,replacing k by 1, 2, . . . , n in turn and considering the resulting system of n homogeneousequations, we get that the resulting matrix of the system is a Van der Monde matrix

⎛

⎜⎜⎜⎜⎜⎜⎝

1 1 ... 12 22 ... 2n

. . ... .

. . ... .

. . ... .

n n2 ... nn

⎞

⎟⎟⎟⎟⎟⎟⎠

.

Since the determinant of the matrix is equal to the product of positive integers, each of which isless then n, and since R is (n+1)!-torsion free, it follows immediately that fi (α(x), β(x), e) =0 for all x ∈ R and i = 1, 2, . . . , n. Now, fn(α(x), β(x), e) = 0 implies that

( n + 1n

)F(x) = F(x) +

( nn − 1

)α(x)F(e) + nD(x)

(n + 1)F(x) = F(x) + nα(x)F(e) + nD(x)

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Algebraic conditions in rings

Since R is n-torsion free, we get

F(x) = α(x)F(e) + D(x) (1.2)

Again, fn−1(α(x), β(x), e) = 0 implies that

( n + 1n − 1

)F(x2) =

( nn − 1

)α(x)F(x) +

( nn − 2

)(α(x))2 F(e)

+ n(n+1)2 β(x)D(x) + n(n−1)

2 α(x)D(x)

n(n + 1)F(x2) = 2nα(x)F(x) + n(n − 1)(α(x))2 F(e) + n(n + 1)β(x)D(x)

+n(n − 1)α(x)D(x) for all x ∈ R.

Since R is n-torsion free, then we obtain

(n + 1)F(x2) = 2α(x)F(x) + (n − 1)(α(x))2 F(e) + (n + 1)β(x)D(x)

+(n − 1)α(x)D(x) for all x ∈ R.

Thanks to (1.2), it becomes

(n+1)F(x2) = (n+1)(α(x))2 F(e)+(n+1)α(x)D(x)+(n+1)β(x)D(x) for all x ∈ R.

Since R is (n + 1)-torsion free, then

F(x2) = (α(x))2 F(e) + α(x)D(x) + β(x)D(x) for all x ∈ R. (1.3)

Replacing x by x2 in (1.2), we get

F(x2) = (α(x))2 F(e) + D(x2) (1.4)

Comparing (1.3) and (1.4), we find that

d(x2) = α(x)D(x) + β(x)D(x) for all x ∈ R. (1.5)

Now, by (1.3), we can write

F(x2) = (α(x))2 F(e) + α(x)D(x) + β(x)D(x)

= α(x)(α(x)F(e) + D(x)) + x D(x) for all x ∈ R.

Again, by (1.2), it follows

F(x2) = α(x)F(x) + β(x)D(x) (1.6)

for all x ∈ R. The relations (1.5) and (1.6) lead to our required conclusions.Following are some immediate results of the above theorem just by taking α = β = IR

(Identity mapping): ��Corollary 1.1 ([5, Theorem 2.1]) Let n ≥ 1 be any fixed integer and let R be any (n + 1)!-torsion free ring. If F, D : R → R are the additive mappings such that F(xn+1) = xn F(x)+nxn D(x) for all x ∈ R, then D is Jordan left derivation and F is generalized Jordan leftderivation on R.

Corollary 1.2 ([5, Corollary 2.2–2.7]) Let n ≥ 1 be any fixed integer and R be any (n +1)!-torsion free semiprime ring. Suppose that F, D : R → R are additive mappings satisfyingF(xn+1) = xn F(x) + nxn D(x) for all x ∈ R. Then the followings are equivalent:

123

A. Z. Ansari, F. Shujat

(i) F is a generalized left derivation on R;(ii) D is a derivation on R and [D(x), y] = 0 for all x, y ∈ R;

(iii) D is a derivation which maps R into Z(R);(iv) R is commutative or D = 0;(v) F(x) = qx for all x ∈ R and some q ∈ Ql(RC ), where Ql(RC ) is left Martindale ring

of quotients;(vi) F is a generalized derivation on R.

Corollary 1.3 Let n ≥ 1 be any fixed integer and R be any (n + 1)!-torsion free ring.Suppose that F, D : R → R are additive mappings satisfying F(xn+1) = (x)n F(x) +∑n

i=1(x)n−i (β(x))i D(x) for all x ∈ R, where n is fixed integer and β is automorphisms ofR. Then, D is Jordan skew derivation and F is generalized Jordan skew derivation on R.

Proof By taking α = IR in Theorem 1.1, we get the required result. ��

In Dhara and Sharma [8], proved a theorem by taking two additive mappings F, d : R → Rsatisfying F(xn+1) = F(x)(x)n + xd(x)(x)n−1 + x2d(x)(x)n−2 + · · · + xnd(x) for allx ∈ R with certain conditions, then F is generalized Jordan derivation and d is Jor-dan derivation on R. Motivated by this result we generalized this theorem by takingtwo additive mappings F, d : R → R with the property F(xn+1) = F(x)(α(x))n +∑n

i=1(β(x))i D(x)(α(x))n−i for all x ∈ R where n is a fixed integer and α, β are auto-morphisms of R. In brief we prove the following:

Theorem 1.2 Let n ≥ 1 be any fixed integer and R be any (n+1)!-torsion free ring. Supposethat F, D : R → R are additive mappings satisfying the relation

F(xn+1) = F(x)(α(x))n +n∑

i=1

(β(x))i D(x)(α(x))n−i for all x ∈ R, (1.7)

where n is fixed integer and α, β are automorphisms of R. Then, D is Jordan (α, β)-derivationand F is generalized Jordan (α, β)-derivation on R.

Proof Since we have given that

F(xn+1) = F(x)(α(x))n +n∑

i=1

(β(x))i D(x)(α(x))n−i for all x ∈ R. (1.8)

Then, replacing x by e, we get nD(e) = 0. Since R is n-torsion free and α(e) = β(e) = e,we obtain D(e) = 0. Again, replacing x by x + ke in (1.8), where k is any positive integer,we get

F((x + ke)n+1) = F(x + ke)(α(x + ke))n

+n∑

i=1

(β(x + ke))i D(x + ke)(α(x + ke))n−i

= (α(x) + ke)n(F(x) + k F(e))

+n∑

i=1

(β(x) + ke))i D(x)(α(x) + ke)n−i for all x ∈ R.

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Algebraic conditions in rings

F[xn+1 +

( n + 11

)xnk +

( n + 12

)xn−1k2 + · · · + kn+1e

]

= (F(x) + k F(e))[(α(x))n +

( n1

)(α(x))n−1k +

( n2

)(α(x))n−2k2 + · · · + kne

]

+n∑

i=1

[[(β(x))i + · · · +

( ii − 2

)(β(x))2ki−2 +

( ii − 1

)(β(x))ki−1 + ki e

]D(x)

(α(x))n−i + · · · +( n − i

n − i − 2

)(α(x))2kn−i−2 +

( n − in − i − 1

)α(x)kn−i−1 + kn−i e

]

for all x ∈ R.

F(xn+1) + F[( n + 1

1

)xnk +

( n + 12

)xn−1k2 + · · · + kn+1e

]

= (α(x))n(F(x) + ke) +[( n

1

)(α(x))n−1k +

( n2

)(α(x))n−2k2 + · · ·

+( n

n − 2

)(α(x))2kn−2 +

( nn − 1

)α(x)kn−1 + kne

](F(x) + k F(e))

+n∑

i=1

[(α(x))n−i +

( n − i1

)(α(x))n−i−1k + · · · +

( n − in − i − 2

)(α(x))2kn−i−2

+( n − i

n − i − 1

)α(x)kn−i−1 + kn−i e

].(β(x))i D(x)

+n∑

i=1

[(α(x))n−i +

( n − i1

)(α(x))n−i−1k + . . . +

( n − in − i − 2

)(α(x))2kn−i−2

+( n − i

n − i − 1

)α(x)kn−i−1 + kn−i e

][( i1

)(β(x))i−1k +

( i2

)(β(x))i−2k2

+ · · · +( i

i − 2

)(β(x))2ki−2 +

( ii − 1

)(β(x))ki−1 + ki e

]D(x)

F[( n + 1

1

)xnk +

( n + 12

)xn−1k2 + · · · + kn+1e

]

= (α(x))nke + (F(x) + k F(e))[( n

1

)(α(x))n−1k +

( n2

)(α(x))n−2k2 + · · ·

+( n

n − 2

)(α(x))2kn−2 +

( nn − 1

)α(x)kn−1 + kne

]

+n∑

i=1(β(x))i D(x)

[( n − i1

)(α(x))n−i−1k + · · ·

+( n − i

n − i − 2

)(α(x))2kn−i−2 +

( n − in − i − 1

)α(x)kn−i−1 + kn−i e

]

+n∑

i=1

[( i1

)(β(x))i−1k +

( i2

)(β(x))i−2k2 + · · · +

( ii − 2

)(β(x))2ki−2

+( i

i − 1

)(β(x))ki−1 + ki e

]D(x).

[(α(x))n−i +

( n − i1

)(α(x))n−i−1k + . . .

+( n − i

n − i − 2

)(α(x))2kn−i−2 +

( n − in − i − 1

)α(x)kn−i−1 + kn−i e

]

This can be written as

k f1(α(x), β(x), e) + k2 f2(α(x), β(x), e) + · · · + kn fn(α(x), β(x), e) = 0

123

A. Z. Ansari, F. Shujat

for all x ∈ R, where fi (α(x), β(x), e) are the coefficients of ki ′s, for all i = 1, 2, . . . , n. Now,replacing k by 1, 2, . . . , n in turn and considering the resulting system of n homogeneousequations, we get that the resulting matrix of the system is a Van der Monde matrix

⎛

⎜⎜⎜⎜⎜⎜⎝

1 1 ... 12 22 ... 2n

. . ... .

. . ... .

. . ... .

n n2 ... nn

⎞

⎟⎟⎟⎟⎟⎟⎠

.

Since the determinant of the matrix is equal to the product of positive integers, each of which isless then n, and since R is (n+1)!-torsion free, it follows immediately that fi (α(x), β(x), e) =0 for all x ∈ R and i = 1, 2, . . . , n. Now, fn(α(x), β(x), e) = 0 implies that

( n + 1n

)F(x) = F(x) +

( nn − 1

)F(e)α(x) + nD(x)

(n + 1)F(x) = F(x) + nF(e)α(x) + nD(x)

Since R is n-torsion free, we get

F(x) = F(e)α(x) + D(x) (1.9)

Again, fn−1(α(x), β(x), e) = 0 implies that

( n + 1n − 1

)F(x2) =

( nn − 1

)F(x)α(x) +

( nn − 2

)F(e)(α(x))2

+ n(n+1)2 β(x)D(x) + n(n−1)

2 D(x)α(x)

n(n + 1)F(x2) = 2nF(x)α(x) + n(n − 1)F(e)(α(x))2 + n(n + 1)β(x)D(x)

+n(n − 1)D(x)α(x) for all x ∈ R.

Since R is n-torsion free, then we obtain

(n + 1)F(x2) = 2F(x)α(x) + (n − 1)F(e)(α(x))2 + (n + 1)β(x)D(x)

+(n − 1)D(x)α(x) for all x ∈ R.

Using (1.9), the above expression turn in to the following;

(n + 1)F(x2) = (n + 1)F(e)(α(x))2 + (n + 1)D(x)α(x)

+(n + 1)β(x)D(x) for all x ∈ R.

Since R is (n + 1)-torsion free, then

F(x2) = F(e)(α(x))2 + α(x)D(x) + D(x)β(x) for all x ∈ R. (1.10)

Replacing x by x2 in (1.9), we get

F(x2) = F(e)(α(x))2 + D(x2) (1.11)

Comparing (1.10) and (1.11), we find that

d(x2) = D(x)α(x) + β(x)D(x) for all x ∈ R. (1.12)

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Algebraic conditions in rings

Now, by (1.10), we can write

F(x2) = F(e)(α(x))2 + D(x)α(x) + β(x)D(x)

= F(e)α(x)(α(x) + D(x)) + x D(x) for all x ∈ R.

Again, by (1.9), it follows

F(x2) = F(x)α(x) + β(x)D(x) for all x ∈ R. (1.13)

Hence, we get the required result. ��Corollary 1.4 Let n ≥ 1 be any fixed integer and R be a (n+1)!-torsion free semiprime ring.

Suppose that D : R → R is an additive mapping satisfying D(xn+1) =n∑

i=0(x)i D(x)(x)n−i

for all x ∈ R. In this case D is a derivation on R.

Proof Taking F = D and α = β = IR in Theorem 1.2, we get D is Jordan derivation. Usingmain theorem of [7], we obtain the required result. ��Corollary 1.5 ([8, Theorem 1]) Let n ≥ 1 be any fixed integer and R be any (n +1)!-torsionfree ring. If F, D : R → R are the additive mappings such that F(xn+1) = F(x)xn +∑n

i=1(x)i D(x)(x)n−i for all x ∈ R, then D is Jordan derivation and F is generalizedJordan derivation on R.

Proof By taking α = β = IR in Theorem 1.2, we get required result. ��

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