Throughout this article, A is a prime ring with center Z(A), right Martindale quotient ring Q, extended centroid C and p∈A, 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:A→A is a skew derivation of A if d(ηω)=d(η)ω+α(η)d(ω) for all η,ω∈A, 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 a∈A, the mapping Ia(η)=[η,a] is derivation called an inner derivation of A. An additive map F:A→A is a generalized skew derivation if F(ηω)=F(η)ω+α(η)d(ω) for all η,ω∈A, where d is an skew derivation of A with associated automorphism α. Further, a generalized skew derivation F:A→A is called X - inner if there exist elements a,b∈Q and an automorphism α of A such that F(η)=aη+α(η)b for all η∈A, otherwise it is outer. Similarly, a skew derivation d:A→A is called X- inner if there is element a∈Q and an automorphism α of A such that d(η)=aη−α(η)a for all η∈A, 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 a∈A such that aηnd(η)m=0 for all η∈L, where n≥1 and m≥1 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 A≠M2(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 η∈L, where s≥0,t≥0 are fixed integers, then H(η)=0, for all η∈A, 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 0≠a∈A. Suppose that aηsH(η)ηt=0 for all η∈L, where s,t≥0 and n≥1 are fixed integers. Then either s = 0 and there exists b∈U such that H(η)=bη for all η∈A 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 p∈A, a nonzero fixed element. If pF(η)η∈C for any η∈L, then A satisfies S₄.

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) dimCAC≤4.

• (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 V_D be a vector space over a division ring D with dimVD≥2 and T∈End(V). If s and Ts are D-dependent for every s∈V, then there exists χ∈D such that Ts=χs for every s∈V.

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,dimDV≥2,f∈End(V) and a∈A. If as=0, for any s∈V such that {s,f(s)} is linearly D-independent, then a=0, unless dimDV=2 and char(A)=2.

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

Now, let r∈V such that {r,s} is linearly D-independent and ar≠0. By the hypothesis, we have {r,f(r)} is linearly D-dependent, as are {r+s,f(r+s)} and {r−s,f(r−s)}. Thus, there exists κr,κr+s,κr−s∈D such that

In other words, we have

and

Suppose dimDV≥3. It is obvious that r∈Span{s,f(s)}, otherwise equation (3.1) is contradicted. Thus for any r∈V, we have r∈Span{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

and

By (3.3) and since {r,s} is D-independent and char(A)≠2, we have κr=κr+s=κr−s. 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 r∈V, i,e. aV=(0). Hence, a=0 follows.

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

Proof. By hypothesis, I satisfies [pa[η1,η2]2,[ω,ω2]]=0 for all η1,η2,ω1,ω2∈I. In particular η1=b, we get [pa[b,η2]2,[ω1,ω2]]=0. Then by [1, Lemma 2.2], we get pa[b,η2]2∈C. 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], dimCAC≤4. Hence, by Fact 2.2, A satisfies S₄.

Proposition 3.3. Let A be a prime ring, a,b∈A,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,ω2∈I. Then A satisfies S₄.

Proof. Suppose b∈C, then by given hyphothesis [p(a+b)[η1,η2]]2,[ω1,ω2]]=0 for all η1,η2,ω1,ω2∈I. Hence, by Lemma 3.2 the required conclusion follows. In case b∉C, assume dimCV≥3 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 s∈V exists such that {s,bs} are linearly C-independent. Since dimCV≥3, then there exists w∈V such that {s,bs,w} are linearly C-independent. By the density of Q, there exists h1,h2,k1,k2∈Q such that

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 s∈V 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 s∈V. Then by Fact 2.5, there exists κ∈C such that bs=κs for any s∈V. For any r∈A, we have that [b,r]s=b(rs)−rbs=κrs−κrs=0, that is, [b,r]V=0. Hence [b,r]= 0 for any r∈A, which implies that b ∈ C. This is a contradiction. Thus, dimCV≤2. Hence, A satisfies S₄.

In this case, we have a,b∈Q such that F(η)=aη+α(η)b for all η∈A, where α∈Aut(Q).

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

for any η1,η2,ω1,ω2∈I. Then A satisfies S₄.

Proof. Suppose q−1b∈C, then by (4.1), we have [p(a+b)[η1,η2]2,[ω1,ω2]]=0, for all η1,η2,ω1,ω2∈I. Hence by Lemma 3.2, A satisfies S₄. Furthermore, in case q∈C, we have [p(a[η1,η2]+[η1,η2]b)[η1,η2],[ω1,ω2]]=0 for any η1,η2,ω1,ω2∈I, and hence by Proposition 3.2, we get the conclusion. From now, we have q−1b∉C and q∉C. 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 dimCV≥3 and suppose that there exists s∈V such that {s,q−1bs} are linearly C-independent. Since dimCV≥3, then there exists w∈V such that {s,q−1bs,w} are linearly C-independent. By the density of Q, there is h1,h2,k1,k2 such that

This gives, 0=([p(a[h1,h2]+q[h1,h2]q−1b)[h1,h2],[k1,k2]])s=pqs. Hence, we have proved pqs=0 for any vector s∈V such that {s,q−1bs} are linearly C-independent. By Lemma 3.1, pq=0 follows a contradiction. Thus, {s,q−1bs} are linearly C-dependent for all s∈V. Then by Fact 2.5, there exists κ∈C such that q−1bs=κs for any s∈V. For any r∈A, we have that [q−1b,r]s=q−1b(rs)−rq−1bs=κrs−κrs=0, that is, [q−1b,r]V=0. Hence [q−1b,r]=0 for any r∈A, which implies that q−1b∈C. This is a contradiction. Thus, dimCV≤2. Hence, A satisfies S₄.

Lemma 4.2. Let A be a prime ring of char(A)≠2, α:A→A be an outer automorphism of A. suppose that a,b∈A such that

for all η1,η2,ω1,ω2∈A. Then A satisfies S₄.

Proof. Assume α≠I, 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 T∈End(V) such that α(η)=TηT−1 for all η∈Q. Assume that s and T−1bs are linearly C-dependent for all s∈V. By Fact 2.5, there exists κ ∈ C such that T−1bs=κs for all s∈V. In this case for all η∈Q, (aη+α(η)b)s=(aη+TηT−1b)s=aηs+TηT−1bs=aηs+T(κηs)=aηs+T(κ(ηs))=aηs+T(T−1b)(ηs)=(a+b)ηs. This means that (aη+α(η)b)s=(a+b)ηs for all η∈Q and s∈V, 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 s∈V such that {s,T−1bs} are linearly C-independent. Assume dimCV≥3, then there exists w∈V such that {s,T−1bs,w} are linearly C-independent. By the density of Q, there exists h1,h2,k1,k2 such that

0=([p(a[h1,h2]+T[h1,h2]T−1b)[h1,h2],[k1,k2]])s=pT(s). Hence, we have proved that pT(s)=0 for every s∈V such that {s,T−1bs} are linearly C-independent. By Lemma 3.1, p = 0 follows a contradiction. Thus dimCV≤2. 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 a∈Q such that F(η)=aη+d(η) for all η∈A.

Case 1: If d is inner, then d(η)=bη−α(η)b for some b∈Q for all η∈Q, So that F(η)=(a+b)η−α(η)b. Then, we have

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

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 q∈Q−C, such that α(η)=qηq−1 for all η∈Q, 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

for all η1,η2,ω1,ω2∈Q.

This implies that

for all η1,η2,ω1,ω2∈Q.

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

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

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