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### Article

Kyungpook Mathematical Journal 2023; 63(3): 325-331

Published online September 30, 2023 https://doi.org/10.5666/KMJ.2023.63.3.325

### Annihilating Conditions of Generalized Skew Derivations on Lie Ideals

Department of Mathematics, Aligarh Muslim University, 202002 Aligarh, India
e-mail : nu.rehman.mm@amu.ac.in and paryamu@gmail.com

Department of Applied Sciences, Symbiosis Institute of Technology, Symbiosis International (Deemed) University, Pune 412115, India
e-mail : junaidnisar73@gmail.com; junaid.nisar@sitpune.edu.in

Received: March 29, 2022; Revised: August 7, 2022; Accepted: August 22, 2022

Let A be a prime ring of char(A)2, L a non-central Lie ideal of A,F a generalized skew derivation of A and pA, a nonzero fixed element. If pF(η)ηC for any ηL, then A satisfies S4.

Keywords: Prime ring, Generalized skew derivation

Throughout this article, A is a prime ring with center Z(A), right Martindale quotient ring Q, extended centroid C and pA, a nonzero fixed element. Any information about definitions and main properties can be found in [3]. The standard polynomial identity S4 in four variables is defined as S4(η1,η2,η4,η4)=(1)σησ(1)ησ(2)ησ(3)ησ(4), where (1)σ is +1 when σ is an even permutation, and (1)σ is (-1), when σ is an odd permutation in the symmetric group S4. An additive map d:AA is a skew derivation of A if d(ηω)=d(η)ω+α(η)d(ω) for all η,ωA, where α is associated automorphism of d. If α is an identity automorphism of A then d is derivation of A. In particular, for a fixed aA, the mapping Ia(η)=[η,a] is derivation called an inner derivation of A. An additive map F:AA is a generalized skew derivation if F(ηω)=F(η)ω+α(η)d(ω) for all η,ωA, where d is an skew derivation of A with associated automorphism α. Further, a generalized skew derivation F:AA is called X - inner if there exist elements a,bQ and an automorphism α of A such that F(η)=aη+α(η)b for all ηA, otherwise it is outer. Similarly, a skew derivation d:AA is called X- inner if there is element aQ and an automorphism α of A such that d(η)=aηα(η)a for all ηA, otherwise, it is X-outer. The notion of a generalized skew derivation is combination of the notions of skew derivation and generalized derivation.

In [13], Sharma and Dhara proved that if A is a prime ring with non-zero derivation d, L a non-central Lie ideal and aA such that aηnd(η)m=0 for all ηL, where n1 and m1 are fixed integers, then one of the following holds:

• (1) a =0 or d(L)=0 if char(A)2.

• (2) a = 0 or d(A)=0 if [L,L]0 and AM2(F).

In [8], Dhara and De Filippis considered generalized derivations. They proved for a prime ring A that if H is a generalized derivation of A and L a non-commutative Lie ideal of A such that ηs H(η) ηt=0 for all ηL, where s0,t0 are fixed integers, then H(η)=0, for all ηA, unless Char(A)=2 and A satisfies S4.

In [9], Du and Wang demonstrated the following result for generalized derivations. Let A be a prime ring, U be its Utumi ring of quotients, H a nonzero generalized derivation of A, L a non-central Lie ideal of A and 0aA. Suppose that aηsH(η)ηt=0 for all ηL, where s,t0 and n1 are fixed integers. Then either s = 0 and there exists bU such that H(η)=bη for all ηA with ab = 0 or A satisfies S4.

Inspired by the above outcomes, in the present paper, we prove the following result about generalized skew derivations with an annihilating condition on the non-central Lie ideal.

Theorem 1.1. Let A be a prime ring of Char(A)2, L a non-central Lie ideal of A,F a generalized skew derivation of A and pA, a nonzero fixed element. If pF(η)ηC for any ηL, then A satisfies S4.

### 2. Preliminaries

The following facts are often referenced in the proofs of our results:

Fact 2.1. Let A be a prime ring and I be a two sided ideal of A. Then I,A and Q satisfy the same generalized polynomial identity with coefficients in Q [5]. Furthermore, I, A and Q satisfy the same generalized polynomial identity with automorphisms [6].

Fact 2.2. ([14, Lemma 2.1]) Let A be a prime ring with extended centroid C. Then the following conditions are equivalent:

• (1) dimCAC4.

• (2) A satisfies S4.

• (3) A is commutative or A embeds in M2(F), for F a field.

• (4) A is algebraic of bounded degree 2 over C.

• (5) A satisfies [[a2,b],[a,b]]=0.

Fact 2.3. ([7, Theorem 1]) Let A be a prime ring, D be an X-outer skew derivation of A and α be an X-outer automorphism of A. If (ψ(ai,),D(ai),α(ai)) is a generalized polynomial identity for A, then A also satisfies the generalized polynomial identity ψ(ai,bi,ci), where ai,bi,ci are distinct indeterminates.

Fact 2.4. Let A be a prime ring and L a be non-central Lie ideal of A. If char(A)2, by [3, Lemma 1] there exists a nonzero ideal I of A such that 0[I,A]L. If char(A)=2 and dimCAC>4, i.e., char(A)=2 and A does not satisfy S4, then by [11, Theorem 13] there exists a nonzero ideal I of A such that 0[I,A]L. Thus, if either char(A)2 or A does not satisfy S4, then we may conclude that there exists a nonzero ideal I of A such that [I,I]L.

Fact 2.5. ([2, Lemma 7.1]) Let VD be a vector space over a division ring D with dimVD2 and TEnd(V). If s and Ts are D-dependent for every sV, then there exists χD such that Ts=χs for every sV.

### 3. Some Important Results

We start with the following lemma and proposition; they are required for the development of our theorems:

Lemma 3.1. Suppose A is a primitive ring isomorphic to a dense ring of linear transformations on some vector space V over a division ring D,dimDV2,fEnd(V) and aA. If as=0, for any sV such that {s,f(s)} is linearly D-independent, then a=0, unless dimDV=2 and char(A)=2.

Proof. A vector sV is fixed such that {s,f(s)} is linearly D-independent, then as=0. Let rV such that {r,s} is linearly D-dependent. Then both ar=0 and r span {s,f(s)} are trivial.

Now, let rV such that {r,s} is linearly D-independent and ar0. By the hypothesis, we have {r,f(r)} is linearly D-dependent, as are {r+s,f(r+s)} and {rs,f(rs)}. Thus, there exists κr,κr+s,κrsD such that

f(r)=rκr,   f(r+s)=(r+s)κr+s,    f(rs)=(rs)κrs.

In other words, we have

rκr+f(s)=rκr+s+sκr+s

and

Suppose dimDV3. It is obvious that rSpan{s,f(s)}, otherwise equation (3.1) is contradicted. Thus for any rV, we have rSpan{s,f(s)}, that is V=Span{s,f(s)}, a contradiction.

To complete the proof, suppose dimDV=2 and suppose char(A)2, if not we are done.

By equating (3.1) and (3.2), we get both

r(2κrκr+sκrs)+s(κrsκr+s)=0

and

2f(s)=r(κr+sκrs)+s(κr+s+κrs).

By (3.3) and since {r,s} is D-independent and char(A)2, we have κr=κr+s=κrs. Thus, 2f(s)=2sκr by (3.1). Since {s,f(s)} is D-independent, the conclusion κr=κr+s=0 follows, that is f(r)=0 and char(A)2, it follows that ar=0, for any rV, i,e. aV=(0). Hence, a=0 follows.

Lemma 3.2. Let A be a non-commutative prime ring, aA,I a nonzero two-sided ideal of A such that [pa[η1,η2]2,[ω1,ω2]]=0 for all η1,η2,ω1,ω2I. Then A satisfies S4.

Proof. By hypothesis, I satisfies [pa[η1,η2]2,[ω,ω2]]=0 for all η1,η2,ω1,ω2I. In particular η1=b, we get [pa[b,η2]2,[ω1,ω2]]=0. Then by [1, Lemma 2.2], we get pa[b,η2]2C. Since [b,η2] is an nonzero inner derivation of A then pa[b,η2]2 is a central DI for I. Thus by [4, Lemma 2], dimCAC4. Hence, by Fact 2.2, A satisfies S4.

Proposition 3.3. Let A be a prime ring, a,bA,I be a nonzero two-sided ideal of A such that [p(a[η1,η2]+[η1,η2]b)[η1,η2],[ω1,ω2]]=0 for any η1,η2,ω1,ω2I. Then A satisfies S4.

Proof. Suppose bC, then by given hyphothesis [p(a+b)[η1,η2]]2,[ω1,ω2]]=0 for all η1,η2,ω1,ω2I. Hence, by Lemma 3.2 the required conclusion follows. In case bC, assume dimCV3 then [p(a[η1,η2]+[η1,η2]b)[η1,η2],[ω1,ω2]]=0 is a non-trivial generalized polynomial identity (GPI) for I. Then by Fact 2.1 [p(a[η1,η2]+[η1,η2]b)[η1,η2],[ω1,ω2]]=0 is a non-trivial GPI for A and Q also. By Martindale Theorem [12], Q is a primitive ring having non-zero socle and its associated division ring is finite dimensional over C. Hence, by Jacobson Theorem in [10, Page 75], Q is isomorphic to a dense ring of linear transformation on some vector space V over C. Assume that sV exists such that {s,bs} are linearly C-independent. Since dimCV3, then there exists wV such that {s,bs,w} are linearly C-independent. By the density of Q, there exists h1,h2,k1,k2Q such that

h1s=s,  h2s=0,  k1w=w,  k2s=w,  k1s=0,
h1w=0,  h2w=s,  h1bs=0,  h2bs=s.

This gives, 0=(p(a[h1,h2]+[h1,h2]b)[h1,h2],[k1,k2]])s=ps. Hence, we have proved ps=0 for any vector sV such that {s,bs} are linearly C-independent. By Lemma 3.1, p = 0 follows a contradiction. Thus, {s,bs} are linearly C-dependent for all sV. Then by Fact 2.5, there exists κC such that bs=κs for any sV. For any rA, we have that [b,r]s=b(rs)rbs=κrsκrs=0, that is, [b,r]V=0. Hence [b,r]= 0 for any rA, which implies that b ∈ C. This is a contradiction. Thus, dimCV2. Hence, A satisfies S4.

### 4. Case of inner generalized skew derivation

In this case, we have a,bQ such that F(η)=aη+α(η)b for all ηA, where αAut(Q).

Lemma 4.1. Let A be a prime ring of Char(A)2 and a,b,qQ. If q is an invertible element of Q and I be a nonzero two-sided ideal of A such that

[p(a[η1η2]+q[η1,η2]q1b)[η1,η2],[ω1,ω2]]=0

for any η1,η2,ω1,ω2I. Then A satisfies S4.

Proof. Suppose q1bC, then by (4.1), we have [p(a+b)[η1,η2]2,[ω1,ω2]]=0, for all η1,η2,ω1,ω2I. Hence by Lemma 3.2, A satisfies S4. Furthermore, in case qC, we have [p(a[η1,η2]+[η1,η2]b)[η1,η2],[ω1,ω2]]=0 for any η1,η2,ω1,ω2I, and hence by Proposition 3.2, we get the conclusion. From now, we have q1bC and qC. So, (4.1) is a non-trivial GPI for I. Then by Fact 2.1, (4.1) is a non-trivial GPI for A and Q also. By Martindale Theorem in [12], Q is a primitive ring having non-zero socle and its associated division ring is finite dimensional over C. Hence, by Jacobson Theorem in [10, page 75], Q is isomorphic to a dense ring of linear transformation on some vector space V over C. Assume dimCV3 and suppose that there exists sV such that {s,q1bs} are linearly C-independent. Since dimCV3, then there exists wV such that {s,q1bs,w} are linearly C-independent. By the density of Q, there is h1,h2,k1,k2 such that

h1s=s,  h2s=0,  k1s=0,  k2s=w,  k1w=w
h1w=0,  h2w=s,  h1q1bs=0,  h2q1bs=s

This gives, 0=([p(a[h1,h2]+q[h1,h2]q1b)[h1,h2],[k1,k2]])s=pqs. Hence, we have proved pqs=0 for any vector sV such that {s,q1bs} are linearly C-independent. By Lemma 3.1, pq=0 follows a contradiction. Thus, {s,q1bs} are linearly C-dependent for all sV. Then by Fact 2.5, there exists κC such that q1bs=κs for any sV. For any rA, we have that [q1b,r]s=q1b(rs)rq1bs=κrsκrs=0, that is, [q1b,r]V=0. Hence [q1b,r]=0 for any rA, which implies that q1bC. This is a contradiction. Thus, dimCV2. Hence, A satisfies S4.

Lemma 4.2. Let A be a prime ring of char(A)2, α:AA be an outer automorphism of A. suppose that a,bA such that

[p(a[η1,η2]+α([η1,η2])b)[η1,η2],[ω1,ω2]]=0

for all η1,η2,ω1,ω2A. Then A satisfies S4.

Proof. Assume αI, otherwise the conclusion directly follows from Proposition 3.3. By Fact 2.1, A is a GPI-ring and Q is also a GPI-ring. By Martindale Theorem in [12], Q is a primitive ring having non-zero socle and its associated division ring is finite dimensional over C. Hence, by Jacobson Theorem in [10][page 79], Q is isomorphic to a dense ring of linear transformation on some vector space V over C. By [13, page 79], there exists a semi-linear transformation TEnd(V) such that α(η)=TηT1 for all ηQ. Assume that s and T1bs are linearly C-dependent for all sV. By Fact 2.5, there exists κ ∈ C such that T1bs=κs for all sV. In this case for all ηQ, (aη+α(η)b)s=(aη+TηT1b)s=aηs+TηT1bs=aηs+T(κηs)=aηs+T(κ(ηs))=aηs+T(T1b)(ηs)=(a+b)ηs. This means that (aη+α(η)b)s=(a+b)ηs for all ηQ and sV, since V is faithful, it follows that (aη+α(η)b)=(a+b)η. Thus, (4.2) reduces to [p(a+b)[η1,η2]2,[ω1,ω2]]=0. Hence, by Lemma 3.1, A satisfies S4. Now, Suppose there exists sV such that {s,T1bs} are linearly C-independent. Assume dimCV3, then there exists wV such that {s,T1bs,w} are linearly C-independent. By the density of Q, there exists h1,h2,k1,k2 such that

h1T1bs=0,   h2T1bs=s,   k1s=0,   k2s=w,   k1w=w
h1s=s,   h2s=0,   h1w=0,   h2w=s

0=([p(a[h1,h2]+T[h1,h2]T1b)[h1,h2],[k1,k2]])s=pT(s). Hence, we have proved that pT(s)=0 for every sV such that {s,T1bs} are linearly C-independent. By Lemma 3.1, p = 0 follows a contradiction. Thus dimCV2. Hence A satisfies S4.

Proof of Theorem 1.1. In view of the Fact 2.4, a nonzero two-sided ideal I exists such that 0[I,A]L. Therefore , I satisfies [pF([η1,η2])[η1,η2],[ω1,ω2]]=0. we known there exists a skew derivation d of A and an element aQ such that F(η)=aη+d(η) for all ηA.

Case 1: If d is inner, then d(η)=bηα(η)b for some bQ for all ηQ, So that F(η)=(a+b)ηα(η)b. Then, we have

[p((a+b)[η1,η2]α([η1,η2])b)[η1,η2],[ω1,ω2]]=0 for all  η1,η2,ω1,ω2I

Subcase 1: When α is an identity map, then the conclusion follows from Proposition 3.3.

Subcase 2: When α is inner , then there is qQC, such that α(η)=qηq1 for all ηQ, then the conclusion follows from Lemma 4.1.

Subcase 3: When α is outer, then the conclusion follows from Lemma 4.2.

Case 2: When d is outer, then we get

[p(a[η1,η2]+d([η1,η2])[η1,η2],[ω1,ω2]]=0

for all η1,η2,ω1,ω2Q.

This implies that

[p(a[η1,η2]+d(η1)η2+α(η1)d(η2)d(η2)η1α(η2)d(η1))[η1,η2],[ω1,ω2]]=0

for all η1,η2,ω1,ω2Q.

Here d is not inner, by applying Fact 2.3, A satisfies

[p(a[η1,η2]+t1η2+α(η1)t2t2η1α(η2)t1)[η1,η2],[ω1,ω2]]=0

In particular, t1=t2=0 , we get [pa[η1,η2]2,[ω1,ω2]]=0, then again by Lemma 3.2, the given conclusion follows.

### Acknowledgement.

The authors are greatly indebted to the referee for his/her constructive comments and suggestions, which improved the quality of the paper.

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