b-Generalized Skew Derivations on Multilinear Polynomials in Prime Rings

Abstract

Let R be a prime ring of characteristic different from 2, Q _r be its right Martindale quotient ring and C be its extended centroid. In this paper we define b-generalized skew derivations of prime rings. Then we describe all possible forms of two b-generalized skew derivations F and G satisfying the condition F(x)x − xG(x) = 0, for all x ∈ S, where S is the set of the evaluations of a multilinear polynomial f(x ₁, …, x _n) over C with n non-commuting variables. Several potential research topics related to our current work are also presented.

1 Introduction

In this paper, unless otherwise mentioned, R always denotes a prime ring with center Z(R). We denote the right Martindale quotient ring of R by Q _r. The center of Q _r is denoted by C, which is called extended centroid of R. We refer the reader to the book [4] for more details.

An additive mapping d: R→R is said to be a derivation of R if

$$\displaystyle \begin{aligned} d(xy)=d(x)y+xd(y) \end{aligned}$$

for all x, y ∈ R. An additive mapping F : R→R is called a generalized derivation of R if there exists a derivation d of R such that

$$\displaystyle \begin{aligned} F(xy)=F(x)y+xd(y) \end{aligned}$$

for all x, y ∈ R. The derivation d is uniquely determined by F, which is called an associated derivation of F.

The definition of generalized skew derivation is a unified notion of skew derivation and generalized derivation, which are considered as classical additive mappings of non-commutative algebras, have been investigated by many people from various views, see [1, 9, 11,12,13,14, 16, 24, 25, 28, 29, 39, 42, 45]. Let R be an associative ring and α be an automorphism of R. An additive mapping d: R→R is said to be a skew derivation of R if

$$\displaystyle \begin{aligned} d(xy)=d(x)y+\alpha(x)d(y) \end{aligned}$$

for all x, y ∈ R. The automorphisms α is called an associated automorphism of d. An additive mapping F : R→R is called a generalized skew derivation of R if there exists a skew derivation d of R with associated automorphism α such that

$$\displaystyle \begin{aligned} F(xy)=F(x)y+\alpha(x)d(y) \end{aligned}$$

for all x, y ∈ R. In this case, d is called an associated skew derivation of F and α is called an associated automorphism of F. It was Chang who first introduced this notion and initiated the study of generalized skew derivations of (semi-)prime rings in [10]. Therein, he described the identity of the form h(x) = af(x) + g(x)b, where f, g and h are the so-called generalized (α, β)-derivations of a prime ring R, a and b are some fixed noncentral elements of R.

It is worth pointing out that many research papers are devoting to studying the additive mappings in the interfaces between algebra and operator algebra. In [7], Brešar and Villena investigate the automatic continuity of skew derivations on Banach algebras and gave the skew derivation version of noncommutative Singer-Wermer conjecture on Banach algebras. Various technical generalizations of derivations on (semi-)prime rings are used to discuss the range inclusion problems of generalized derivations on noncommutative Banach algebras, see [5, 8, 27, 46, 47]. More recently, Eremita et al determine the structure of generalized skew derivations implemented by elementary operators [30]. Liu and his students characterize a (generalized-)skew derivation F of Banach algebras so that the values of F on a left ideal are nilpotent [41, 43]. Qi and Hou in [45] study generalized skew derivations on nest algebras determined by acting on zero products.

Brešar in [6] gives a description of additive mappings which are commuting on a prime ring R. More precisely, he proves that if F is an additive mapping of R into itself which is centralizing on R and if either R has a characteristic different from 2 or F is commuting on R, then F is of the form F(x) = λx + ζ(x), where λ is an element of the extended centroid C of R and ζ is an additive mapping of R into C. Moreover, the general situation when two additive mappings F and G of the ring R satisfy F(x)x − xG(x) ∈ Z(R) for all x in a subset S of R is considered. In particular, it is showed that if 0 ≠ F and G are both derivations of R and S is a nonzero left ideal of R, then R is commutative. Many researchers successfully extended this result concerning derivations, by replacing S with other subsets of R or replacing F and G with other types of additive mappings. In [49], Wong characterizes derivations F and G of R such that F(x)x − xG(x) ∈ Z(R), for all x ∈ S, where S is the set of all the evaluations (in a non-zero ideal of R) of a non-central multilinear polynomial over C. Later, Lee and Shiue in [36] extend Wong’s result to derivations acting on arbitrary polynomials. Then, in [40], Liu generalizes the theorem of Wong to one-sided ideals. More recently, Chen in [15] extends Lee and Shiue’s result to generalized derivations.

In a recent paper [34], Koşan and Lee propose the following new definition. Let d: R→Q _r be an additive mapping and b ∈ Q _r. An additive map F : R→Q _r is called a left b-generalized derivation, with associated mapping d, if F(xy) = F(x)y + bxd(y), for all x, y ∈ R. In the present paper this mapping F will be called b-generalized derivation with associated pair (b, d). Clearly, any generalized derivation with associated derivation d is a b-generalized derivation with associated pair (1, d).

In view of this idea, we now give the following:

Definition 1

Let b ∈ Q _r, d : R→Q _r an additive mapping and α be an automorphism of R. An additive mapping F : R→Q _r is called a b-generalized skew derivation of R, with associated term (b, α, d) if

$$\displaystyle \begin{aligned} F(xy)=F(x)y+b\alpha(x)d(y) \end{aligned}$$

for all x, y ∈ R.

According to the above definition, we can conclude that general results about b-generalized skew derivations may give useful and powerful corollaries about derivations, generalized derivations, skew derivations and generalized skew derivations.

The main goal of the present paper is to prove the following theorem. It characterizes b-generalized skew derivations which are commuting on multilinear polynomials in prime rings:

Theorem 1

Let R be a prime ring of characteristic different from 2, Q _r be its right Martindale quotient ring and C be its extended centroid, α ∈ Aut(R), d and δ skew derivations of R with associated automorphism α, such that both d and δ are commuting with α. Suppose that F, G are b-generalized skew derivations of R, with associated terms (b, α, d) and (p, α, δ), respectively. Let f(x ₁, …, x _n) be a non-central multilinear polynomial over C with n non-commuting variables. If

$$\displaystyle \begin{aligned} F(f(r_1,\ldots, r_n))f(r_1, \ldots, r_n)-f(r_1,\ldots,r_n)G(f(r_1,\ldots, r_n))=0 \end{aligned} $$

(1)

for all r ₁, …, r _n ∈ R, then one of the following statements holds:

• 1. there exists a′∈ Q _r such that, F(x) = xa′ and G(x) = a′x for all x ∈ R;

• 2. f(x ₁, …, x _n)² is central-valued on R and there exist a′, b′∈ Q _r such that F(x) = a′x + xb′, G(x) = b′x + xa′, for all x ∈ R.

Let us recall some results which will be useful in the sequel.

Note 1

Let R be a prime ring, then the following statements hold:

• 1. Every generalized derivation of R can be uniquely extended to Q _r [35, Theorem 3].

• 2. Any automorphism of R can be uniquely extended to Q _r [19, Fact 2].

• 3. Every generalized skew derivation of R can be uniquely extended to Q _r [10, Lemma 2].

Lemma 1

Let R be a prime ring, α ∈Aut(R), 0 ≠ b ∈ Q _r , d: R→R be an additive mapping of R and F be the b-generalized skew derivation of R with associated term (b, α, d). Then d is a skew derivation of R with associated automorphism α.

Proof

See [26, Lemma 3.2].

Lemma 2

Let R be a prime ring, α ∈ Aut(R), b ∈ Q _r , d: R→R be an additive mapping of R and F be the b-generalized skew derivation of R with associated term (b, α, d). Then F can be uniquely extended to Q _r and assumes the form F(x) = ax + bd(x), where a ∈ Q _r.

Proof

See [26, Lemma 3.3].

2 Some Results on Differential Identities with Automorphisms

In order to proceed with our proofs, we need to recall some well-known results on skew derivations and automorphisms involved in generalized polynomial identities for prime rings.

Let us denote by SDer(Q _r) the set of all skew-derivations of Q _r. By a skew-derivation word we mean an additive mapping Δ of the form Δ = d ₁ d ₁…d _m, where d _i ∈SDer(Q _r). A skew-differential polynomial is a generalized polynomial with coefficients in Q _r of the form Φ(Δ _j(x _i)) involving noncommutative indeterminates x _i on which the skew derivation words Δ _j act as unary operations. The skew-differential polynomial Φ(Δ _j(x _i)) is said to be a skew-differential identity on a subset T of Q _r if it vanishes on any assignment of values from T to its indeterminates x _i.

Let R be a prime ring, SD_int be the C-subspace of SDer(Q _r) consisting of all inner skew-derivations of Q _r, and let d and δ be two non-zero skew-derivations of Q _r. The following results follow as special cases from results in [18,19,20,21, 33].

Note 2

Let d and δ be skew derivations on R, associated with the same automorphism α of R. Assume that d and δ are C −linearly independent modulo SD_int.

If d and δ are commuting with the automorphism α and Φ(Δ _j(x _i)) is a skew-differential identity on R, where Δ _j are skew-derivations words from the set {d, δ}, then Φ(y _ji) is a generalized polynomial identity of R, where y _ji are distinct indeterminates (see [33, Theorem 6.5.9]).

In particular, we have

Note 3

In [22] Chuang and Lee investigate polynomial identities with a single skew derivation. They prove that if Φ(x _i, D(x _i)) is a generalized polynomial identity for R, where R is a prime ring and D is an outer skew derivation of R, then R also satisfies the generalized polynomial identity Φ(x _i, y _i), where x _i and y _i are distinct indeterminates. Furthermore, they observe [22, Theorem 1] that in the case Φ(x _i, D(x _i), α(x _i)) is a generalized polynomial identity for a prime ring R, D is an outer skew derivation of R and α is an outer automorphism of R, then R also satisfies the generalized polynomial identity Φ(x _i, y _i, z _i), where x _i, y _i, and z _i are distinct indeterminates.

Note 4

If d and δ are C −linearly dependent modulo SD_int, then there exist λ, μ ∈ C, a ∈ Q _r and α ∈Aut(Q _r) such that λd(x) + μδ(x) = ax − α(x)a for all x ∈ R.

Note 5

By Chuang and Lee [22] we can state the following result. If d is a non-zero skew-derivation of R and

$$\displaystyle \begin{aligned} \varPhi\bigg(x_1,\ldots,x_n,d(x_1),\ldots,d(x_n)\bigg) \end{aligned}$$

is a skew-differential polynomial identity of R, then one of the following statements holds:

• 1. either d ∈SD_int ;

• 2. or R satisfies the generalized polynomial identity

  $$\displaystyle \begin{aligned}\varPhi(x_1, \ldots, x_n, y_1, \ldots,y_n).\end{aligned}$$

Note 6

Let R be a prime ring and I be a two-sided ideal of R. Then I, R, and Q _r satisfy the same generalized polynomial identities with coefficients in Q _r (see [18]). Furthermore, I, R, and Q _r satisfy the same generalized polynomial identities with automorphisms (see [20, Theorem 1]).

Note 7

Let R be a prime ring, Inn(Q _r) be the C-subspace of Aut(Q _r) consisting of all inner automorphisms of Q _r and let α and β be two non-trivial automorphisms of Q _r.

α and β are called mutually outer if αβ ⁻¹ is not an inner automorphism of Q _r.

If α and β are mutually outer automorphisms of Q _r and Φ(x _i, α(x _i), β(x _i)) is an automorphic identity for R, then by Kharchenko [32, Theorem 4] we know that Φ(x _i, y _i, z _i) is a generalized polynomial identity for R, where x _i, y _i, z _i are distinct indeterminates.

Note 8

Let R be a prime ring, α, β ∈Aut(Q _r) and d : R → R be a skew derivation, associated with the automorphism α. If there exist 0 ≠ θ ∈ C, 0 ≠ η ∈ C and u, b ∈ Q _r such that

$$\displaystyle \begin{aligned} d(x)=\theta\bigg(ux-\alpha(x)u\bigg)+\eta\bigg(bx-\beta(x)b\bigg),\quad \forall x\in R \end{aligned} $$

(2)

then d is an inner skew derivation of R. More precisely, either b = 0 or α = β.

Proof

Starting from relation (2) we have

$$\displaystyle \begin{aligned} d(xy)=\theta\bigg(uxy-\alpha(x)\alpha(y)u\bigg)+\eta\bigg(bxy-\beta(x)\beta(y)b\bigg),\quad \forall x,y\in R. \end{aligned} $$

(3)

On the other hand,

$$\displaystyle \begin{aligned} \begin{array}{ll} & d(xy) = d(x)y+\alpha(x)d(y)= \\ & \theta\bigg(ux-\alpha(x)u\bigg)y+\eta\bigg(bx-\beta(x)b\bigg)y+\\ & \alpha(x)\theta\bigg(uy-\alpha(y)u\bigg)+\alpha(x)\eta\bigg(by-\beta(y)b\bigg). \end{array} \end{aligned} $$

(4)

Comparision of (3) with (4) leads to

$$\displaystyle \begin{aligned} \eta\bigg( \beta(x)\beta(y)b-\beta(x)by+\alpha(x)by-\alpha(x)\beta(y)b\bigg)=0,\quad \forall x,y\in R. \end{aligned} $$

(5)

Suppose first that α and β are mutually outer, in the sense of Note 7. Therefore, by (5) and since η ≠ 0, it follows that

$$\displaystyle \begin{aligned} y_1y_2b-y_1by+x_1by-x_1y_2b=0,\quad \forall x,y,x_1,y_1,y_2\in R. \end{aligned} $$

(6)

In particular, for y ₂ = x ₁ = 0 we get y ₁ by = 0, for any y, y ₁ ∈ R and, by the primeness of R, it follows b = 0, as required.

Now we assume that α and β are not mutually outer, that is there exists an invertible element q ∈ Q _r such that αβ ⁻¹(x) = qxq ⁻¹, for any x ∈ R. Replacing x by β(x), it follows easily that α(x) = qβ(x)q ⁻¹. Hence by (5)

$$\displaystyle \begin{aligned} \beta(x)\beta(y)b-\beta(x)by+q\beta(x)q^{-1}by-q\beta(x)q^{-1}\beta(y)b=0,\quad \forall x,y\in R \end{aligned}$$

that is

$$\displaystyle \begin{aligned} \bigg(q\beta(x)q^{-1}-\beta(x)\bigg)\bigg(\beta(y)b-by\bigg)=0,\quad \forall x,y\in R. \end{aligned} $$

(7)

Now replace y by yz in (7), then

$$\displaystyle \begin{aligned} \bigg(q\beta(x)q^{-1}-\beta(x)\bigg)\bigg(\beta(y)\beta(z)b-byz\bigg)=0,\quad \forall x,y,z\in R \end{aligned} $$

(8)

and using (7) in (8) it follows

$$\displaystyle \begin{aligned} \bigg(q\beta(x)q^{-1}-\beta(x)\bigg)\beta(y)\bigg(\beta(z)b-bz\bigg)=0,\quad \forall x,y,z\in R. \end{aligned} $$

(9)

By the primeness of R, one has that either β(z)b − bz = 0, for any z ∈ R, or qβ(x)q ⁻¹ − β(x) = 0, for any x ∈ R. In the first case d(x) = θ(ux − α(x)u) and we are done. In the latter case, for any x ∈ R we get β(x) = qβ(x)q ⁻¹ = α(x) and we are done again.

Note 9

Assuming that f(x ₁, …, x _n) is a multilinear polynomial over C and d is a skew derivation of R, associated with the automorphism α, we denote

$$\displaystyle \begin{aligned}f(x_1,\ldots,x_n)=\sum_{\sigma \in S_n} \gamma_\sigma x_{\sigma(1)}\cdot x_{\sigma(2)}\cdots x_{\sigma(n)},\quad \gamma_\sigma \in C.\end{aligned}$$

Let f ^d(x ₁, …, x _n) be the polynomial originated from f(x ₁, …, x _n) by replacing each coefficient γ _σ with d(γ _σ). Thus

$$\displaystyle \begin{aligned} d\bigg(\gamma_\sigma\cdot x_{\sigma(1)}\cdot x_{\sigma(2)}\cdots x_{\sigma(n)}\bigg)=d(\gamma_\sigma)x_{\sigma(1)}\cdot x_{\sigma(2)}\cdots x_{\sigma(n)}+ \end{aligned}$$

$$\displaystyle \begin{aligned} +\alpha(\gamma_\sigma)\sum_{j=0}^{n-1}\alpha(x_{\sigma(1)}\cdot x_{\sigma(2)}\cdots x_{\sigma(j)})d(x_{\sigma(j+1)})x_{\sigma(j+2)}\cdots x_{\sigma(n)} \end{aligned}$$

and

$$\displaystyle \begin{aligned}d(f(x_1,\ldots,x_n))=f^d(x_1,\ldots,x_n)+\end{aligned}$$

$$\displaystyle \begin{aligned}+\sum_{\sigma \in S_n} \alpha(\gamma_\sigma)\sum_{j=0}^{n-1}\alpha(x_{\sigma(1)}\cdot x_{\sigma(2)}\cdots x_{\sigma(j)})d(x_{\sigma(j+1)})x_{\sigma(j+2)}\cdots x_{\sigma(n)}.\end{aligned}$$

3 Commuting Generalized Derivations and Commuting Generalized Skew Derivations

Here we would like also to collect some results in literature concerning commuting generalized derivations and commuting generalized skew derivations. This section will be useful in the sequel in order to conclude the proof of our main results.

Proposition 1 ([2, Lemma 3])

Let R be a prime ring, Q _r be its right Martindale quotient ring and C be its extended centroid, f(x ₁, …, x _n) be a multilinear polynomial over C, which is not central-valued on R. Suppose there exist a, b, c, q ∈ Q _r such that

$$\displaystyle \begin{aligned} \begin{array}{ll} & \bigg(af(r_1,\ldots,r_n)+f(r_1,\ldots,r_n)b\bigg)f(r_1,\ldots,r_n)\\ & -f(r_1,\ldots,r_n)\bigg(cf(r_1,\ldots,r_n)+f(r_1,\ldots,r_n)q\bigg)=0 \end{array} \end{aligned} $$

(10)

for all r ₁, …, r _n ∈ R. Then one of the following statements holds:

• 1. a, q ∈ C, q − a = b − c = α ∈ C;

• 2. f(x ₁, …, x _n)² is central-valued on R and there exists α ∈ C such that q − a = b − c = α;

• 3. char(R) = 2 and R satisfies S ₄.

Corollary 1

Let R be a prime ring and f(x ₁, …, x _n) be a multilinear polynomial over C with n non-commuting variables. Let a, b ∈ R be such that

$$\displaystyle \begin{aligned} af(r_1,\ldots,r_n)^2+f(r_1,\ldots,r_n)bf(r_1,\ldots,r_n)=0 \end{aligned}$$

for all r ₁, …, r _n ∈ R. If f(x ₁, …, x _n) is not central valued on R, then either a = −b ∈ C, or char(R) = 2 and R satisfies S ₄.

Lemma 3 ([2, Lemma 1])

Let R be a prime ring and f(x ₁, …, x _n) be a polynomial over C with n non-commuting variables. Let a, b ∈ R be such that af(r ₁, …, r _n) + f(r ₁, …, r _n)b = 0 for all r ₁, …, r _n ∈ R. If f(x ₁, …, x _n) is not a polynomial identity for R, then either a = −b ∈ C, or f(x ₁, …, x _n) is central-valued on R and a + b = 0, unless char(R) = 2 and R ⊆ M ₂(C), the 2 × 2 matrix ring over C.

Corollary 2

Let R be a prime ring of characteristic different from 2 and f(x ₁, …, x _n) be a polynomial over C with n non-commuting variables. Let a ∈ R be such that f(r ₁, …, r _n)a = 0 (or af(r ₁, …, r _n) = 0) for all r ₁, …, r _n ∈ R. If f(x ₁, …, x _n) is not a polynomial identity for R, then a = 0.

Theorem 2 ([2, Theorem 1])

Let R be a prime ring, Q _r be its right Martindale quotient ring and C be its extended centroid, I a non-zero two-sided ideal of R, F and G non-zero generalized derivations of R. Suppose that f(x ₁, …, x _n) is a non-central multilinear polynomial over C such that

$$\displaystyle \begin{aligned}F(f(r_1,\ldots,r_n))f(r_1,\ldots,r_n)-f(r_1,\ldots,r_n)G(f(r_1,\ldots,r_n))=0\end{aligned}$$

for all r ₁, …, r _n ∈ I, then one of the following statements holds:

• 1. there exists a ∈ Q _r such that, F(x) = xa and G(x) = ax for all x ∈ R;

• 2. f(x ₁, …, x _n)² is central-valued on R and there exist a, b ∈ Q _r such that F(x) = ax + xb, G(x) = bx + xa, for all x ∈ R;

• 3. char(R) = 2 and R satisfies S ₄ , the standard identity of degree 4.

4 Some Remarks on Matrix Algebras

Let us state some well-known facts concerning the case when R = M _m(K) is the algebra of m × m matrices over a field K. Note that the set f(R) = {f(r ₁, …, r _n)|r ₁, …, r _n ∈ R} is invariant under the action of all inner automorphisms of R. Let us write r = (r ₁, …, r _n) ∈ R × R ×… × R = R ⁿ. Then for any inner automorphism φ of M _m(K), we get that $\underline {r}=(\varphi (r_1), \ldots , \varphi (r_n))\in R^n$ and $\varphi (f(r))=f( \underline {r})\in f(R)$. As usual, we denote the matrix unit having 1 in (i, j)-entry and zero elsewhere by e _ij.

Let us recall some results from [37]. Let T be a ring with 1 and let e _ij ∈ M _m(T) be the matrix unit having 1 in (i, j)-entry and zero elsewhere. For a sequence u = (A ₁, …, A _n) in M _m(T), the value of u is defined to be the product |u| = A ₁ A ₂⋯A _n and u is nonvanishing if |u|≠0. For a permutation σ of {1, 2, ⋯ , n}, we write u ^σ = (A _σ(1), …, A _σ(n)). We call u simple if it is of the form $u=(a_1e_{i_1j_1},\cdots ,a_ne_{i_nj_n})$, where a _i ∈ T. A simple sequence u is called even if for some σ, |u ^σ| = be _ii≠0, and odd if for some σ, |u ^σ| = be _ij≠0, where i≠j. In [37] it is proved that:

Note 10

Let T be a K-algebra with 1 and let R = M _m(T), m ≥ 2. Suppose that g(x ₁, …, x _n) is a multilinear polynomial over K such that g(u) = 0 for all odd simple sequences u. Then g(x ₁, …, x _n) is central-valued on R.

Note 11

Let T be a K-algebra with 1 and let R = M _m(T), m ≥ 2. Suppose that g(x ₁, …, x _n) is a multilinear polynomial over K. Let u = (A ₁, …, A _n) be a simple sequence from R.

• 1. If u is even, then g(u) is a diagonal matrix.

• 2. If u is odd, then g(u) = ae _pq for some a ∈ T and p ≠ q.

We also notice that:

Note 12

Since f(x ₁, …, x _n) is not central-valued on R, then by Note 10 there exists an odd simple sequence r = (r ₁, …, r _n) from R such that f(r) = f(r ₁, …, r _n) ≠ 0. By Note 11, f(r) = βe _pq, where 0≠β ∈ C and p ≠ q. Since f(x ₁, …, x _n) is a multilinear polynomial and C is a field, we may assume that β = 1. Now, for distinct i, j, let σ ∈ S _n be such that σ(p) = i and σ(q) = j, and let ψ be the automorphism of R defined by ψ(∑_s,t ξ _st e _st) =∑_s,t ξ _st e _σ(s)σ(t). Then f(ψ(r)) = f(ψ(r ₁), …, ψ(r _n)) = ψ(f(r)) = βe _ij = e _ij.

Note 13

By Note 11 and [37, Lemma 9], since f(x ₁, …, x _n) is not central-valued on R, then there exists a sequence of matrices r ₁, …, r _n ∈ R such that f(r ₁, …, r _n) =∑_i α _i e _ii = D is a non-central diagonal matrix, for α _i ∈ C. Suppose r ≠ s such that α _r ≠ α _s. For all l ≠ m, let ψ ∈Aut_C(R) defined by ψ(x) = ψ(∑_ij α _ij e _ij) =∑_ij α _ij e _σ(i)σ(j), where σ is a permutation in the symmetric group of n elements, such that σ(r) = l and σ(s) = m. Thus ψ(D) is an element of f(R) and it is a diagonal matrix with (l, l) and (m, m) entries distinct.

Note 14 ([23, Lemma 1.5])

Let H be an infinite field and n ≥ 2. If A ₁, …, A _k are not scalar matrices in $M_m(\mathcal {H})$ then there exists some invertible matrix P ∈ M _m(H) such that each matrix PA ₁ P ⁻¹, …, PA _k P ⁻¹ has all non-zero entries.

5 Commuting Inner b-Generalized Skew Derivations

The present section is devoted to the proof of a reduced version of Theorem 1. More precisely, we prove the Theorem in the case α, β are automorphisms of R and F, G are inner b-generalized skew derivations of R respectively defined as follows:

$$\displaystyle \begin{aligned}F(x)=ax+b\alpha(x)c,\qquad G(x)=ux+p\beta(x)w\end{aligned}$$

for all x ∈ R and suitable fixed a, b, c, u, p, w ∈ Q _r.

We would like to remark that in this section F and G have not necessarily the same associated automorphism.

We start with the following case:

Lemma 4

Let R = M _m(C), m ≥ 2 and let C be infinite. Suppose that F, G are inner b-generalized skew derivations of R respectively defined as follows:

$$\displaystyle \begin{aligned}F(x)=ax+bqxq^{-1}c,\qquad G(x)=ux+pvxv^{-1}w\end{aligned}$$

for all x ∈ R and suitable fixed a, b, c, u, p, w, q, v ∈ Q _r , with invertible elements q, v of Q _r . Let f(x ₁, …, x _n) be a non-central multilinear polynomial over C with n non-commuting variables. If

$$\displaystyle \begin{aligned} F(f(r_1,\ldots, r_n))f(r_1, \ldots, r_n)-f(r_1,\ldots,r_n)G(f(r_1,\ldots, r_n))=0 \end{aligned} $$

(11)

for all r ₁, …, r _n ∈ R, then the following statements hold simultaneously:

• 1. either bq ∈ Z(R) or q ⁻¹ c ∈ Z(R).

• 2. either pv ∈ Z(R) or v ⁻¹ w ∈ Z(R).

Proof

We assume that bq∉Z(R) and q ⁻¹ c∉Z(R), that is both q ⁻¹ c and bq are not scalar matrices, and prove that a contradiction follows. By Note 14, there exists some invertible matrix P ∈ M _m(C) such that each matrix PbqP ⁻¹, P(q ⁻¹ c)P ⁻¹ has all non-zero entries. Denote by φ(x) = PxP ⁻¹ the inner automorphism induced by P. Say φ(bq) =∑_hl q _hl e _hl and φ(q ⁻¹ c) =∑_hl c _hl e _hl for 0 ≠ q _hl, 0 ≠ c _hl ∈ C. Without loss of generality, we may replace bq and q ⁻¹ c with φ(bq) and φ(q ⁻¹ c), respectively. Hence, for f(r ₁, …, r _n) = λe _ij ≠ 0 in (11), we get that the (j, j)-entry in (11) is

$$\displaystyle \begin{aligned} q_{ji}c_{ji}=0, \end{aligned}$$

which is a contradiction.

Assume now that pv∉Z(R) and v ⁻¹ w∉Z(R), that is both v ⁻¹ w and pv are not scalar matrices, and prove that a contradiction follows. As above, there exists χ(x) = QxQ ⁻¹ the inner automorphism induced by Q ∈ R, such that χ(pv) =∑_hl p _hl e _hl and χ(v ⁻¹ w) =∑_hl w _hl e _hl for 0 ≠ p _hl, 0 ≠ w _hl ∈ C. Moreover we replace pv and v ⁻¹ w with χ(pv) and χ(v ⁻¹ w), respectively. Hence, again for f(r ₁, …, r _n) = λe _ij ≠ 0 in (20), we observe that the (i, i)-entry in (11) is

$$\displaystyle \begin{aligned} p_{ji}w_{ji}=0, \end{aligned}$$

which is also a contradiction.

Lemma 5

Let R = M _m(C), m ≥ 2 and let char(C) ≠ 2. Suppose that F, G are inner b-generalized skew derivations of R respectively defined as follows:

$$\displaystyle \begin{aligned}F(x)=ax+bqxq^{-1}c,\qquad G(x)=ux+pvxv^{-1}w\end{aligned}$$

for all x ∈ R and suitable fixed a, b, c, u, p, w, q, v ∈ Q _r , with invertible elements q, v of Q _r . Let f(x ₁, …, x _n) be a non-central multilinear polynomial over C with n non-commuting variables. If

$$\displaystyle \begin{aligned} F(f(r_1,\ldots, r_n))f(r_1, \ldots, r_n)-f(r_1,\ldots,r_n)G(f(r_1,\ldots, r_n))=0 \end{aligned}$$

for all r ₁, …, r _n ∈ R, then one of the following assertions holds:

• 1. bq ∈ Z(R) and pv ∈ Z(R);

• 2. bq ∈ Z(R) and v ⁻¹ w ∈ Z(R);

• 3. q ⁻¹ c ∈ Z(R) and pv ∈ Z(R);

• 4. q ⁻¹ c ∈ Z(R) and v ⁻¹ w ∈ Z(R).

Proof

If one assumes that C is infinite, the conclusion follows from Lemma 4.

Now let E be an infinite field which is an extension of the field C and let $\overline {R}=M_t(E)\cong R\otimes _C E$. Notice that the multilinear polynomial f(x ₁, …, x _n) is central-valued on R if and only if it is central-valued on $\overline {R}$. Consider the generalized polynomial

$$\displaystyle \begin{aligned} \begin{array}{ll} & \varPsi(x_1,\ldots,x_{n})=\\ & \bigg(af(x_1,\ldots, x_n)+ bqf(x_1,\ldots,x_n)q^{-1}c\bigg)f(x_1,\ldots,x_n)\\ & - f(x_1,\ldots,x_n)\bigg(uf(x_1,\ldots, x_n)+pvf(x_1, \ldots,x_n)v^{-1}w\bigg), \end{array} \end{aligned} $$

(12)

which is a generalized polynomial identity for R. Moreover, it is multi-homogeneous of multi-degree (2, …, 2) in the indeterminates x ₁, …, x _n. Hence the complete linearization of Ψ(x ₁, …, x _n) is a multilinear generalized polynomial Θ(x ₁, …, x _n, y ₁, …, y _n). Moreover,

$$\displaystyle \begin{aligned} \varTheta(x_1,\ldots,x_{n},x_1,\ldots,x_{n})=2^{n} \varPsi(x_1,\ldots,x_{n}).\end{aligned}$$

Clearly, the multilinear polynomial Θ(x ₁, …, x _n, y ₁, …, y _n) is a generalized polynomial identity for R and $\overline {R}$ too. Since char(C) ≠ 2, we obtain Ψ(r ₁, …, r _n) = 0 for all $r_1,\ldots ,r_{n} \in \overline {R}$, and the conclusion follows from Lemma 4.

Lemma 6

Assume that

$$\displaystyle \begin{aligned} \begin{array}{ll} & \varPsi(x_1,\ldots,x_{n}) = \\ & \bigg(af(x_1,\ldots, x_n)+bqf(x_1,\ldots,x_n)q^{-1}c\bigg)f(x_1,\ldots,x_n)\\ & -f(x_1,\ldots,x_n)\bigg(uf(x_1,\ldots, x_n)+pvf(x_1, \ldots,x_n)v^{-1}w\bigg) \end{array} \end{aligned} $$

(13)

is a generalized polynomial identity for R. If R does not satisfy any non-trivial generalized polynomial identity, then one of the following holds:

• 1. bq ∈ C and p = 0;

• 2. bq ∈ C and v ⁻¹ w ∈ C;

• 3. q ⁻¹ c ∈ C and p = 0;

• 4. q ⁻¹ c ∈ C and v ⁻¹ w ∈ C;

• 5. a = u ∈ C, q ⁻¹ c ∈ C, pv ∈ C, bc = 0 and pw = 0.

Proof

We firstly assume that a∉C.

If {a, bq, 1} is linearly C-independent and since Ψ(x ₁, …, x _n) is a trivial generalized polynomial identity for R, then the component af(x ₁, …, x _n)² is also a trivial generalized identity for R, implying the contradiction a = 0. Hence we assume there exist α, γ ∈ C, such that bq = αa + γ. In this case (13) reduces to

$$\displaystyle \begin{aligned} \begin{array}{ll} & af(x_1,\ldots,x_n)^2+(\alpha a+\gamma)f(x_1,\ldots,x_n)q^{-1}cf(x_1,\ldots,x_n)\\ & -f(x_1,\ldots,x_n)uf(x_1,\ldots,x_n)-f(x_1,\ldots,x_n)pvf(x_1,\ldots,x_n)v^{-1}w. \end{array} \end{aligned} $$

(14)

Since {1, a} is linearly C-independent and (14) is a trivial generalized polynomial identity for R, then the components

$$\displaystyle \begin{aligned} af(x_1,\ldots,x_n)(1+\alpha q^{-1}c) \end{aligned} $$

(15)

and

$$\displaystyle \begin{aligned} \begin{array}{ll} & \gamma f(x_1,\ldots,x_n)q^{-1}cf(x_1,\ldots,x_n)-f(x_1,\ldots,x_n)uf(x_1,\ldots,x_n)\\ & -f(x_1,\ldots,x_n)pvf(x_1,\ldots,x_n)v^{-1}w \end{array} \end{aligned} $$

(16)

are also trivial generalized polynomial identities for R. By (15), we get q ⁻¹ c ∈ C. Thus, in the case v ⁻¹ w ∈ C we are done. Here we assume that v ⁻¹ w∉C, that is {1, v ⁻¹ w} is linearly C-independent. Therefore, by (16) it follows that R satisfies f(x ₁, …, x _n)pvf(x ₁, …, x _n)v ⁻¹ w, which implies pv = 0, that is p = 0 (since v is invertible).

Assume now both a ∈ C and bq ∈ C. Hence (13) reduces to

$$\displaystyle \begin{aligned} \begin{array}{ll} & f(x_1,\ldots, x_n)(a+bc)f(x_1,\ldots,x_n)\\ & -f(x_1,\ldots,x_n)uf(x_1,\ldots, x_n)-f(x_1,\ldots,x_n)pvf(x_1, \ldots,x_n)v^{-1}w. \end{array} \end{aligned} $$

(17)

Also in this case, if v ⁻¹ w ∈ C we are done.

Assume that {1, v ⁻¹ w} is linearly C-independent. Starting from (17) one has that the component f(x ₁, …, x _n)pvf(x ₁, …, x _n)v ⁻¹ w must be a trivial generalized polynomial identity for R. This gives that pv = 0, that is p = 0.

Finally, we consider the case a ∈ C and bq∉C. Thus, by (13) we have that

$$\displaystyle \begin{aligned} \begin{array}{ll} & bq f(x_1,\ldots, x_n)q^{-1}cf(x_1,\ldots,x_n)\\ & +f(x_1,\ldots,x_n)(a-u)f(x_1,\ldots, x_n)-f(x_1,\ldots,x_n)pvf(x_1, \ldots,x_n)v^{-1}w \end{array} \end{aligned} $$

(18)

is a trivial generalized polynomial identity for R. Since bq∉C and by (18), it follows that bqf(x ₁, …, x _n)q ⁻¹ cf(x ₁, …, x _n) is also a trivial generalized polynomial identity for R, implying q ⁻¹ c ∈ C and bc = 0. As above, if v ⁻¹ w ∈ C we are done. On the other hand, if v ⁻¹ w∉C and again by (18), one has that f(x ₁, …, x _n)pvf(x ₁, …, x _n)v ⁻¹ w is a trivial generalized polynomial identity for R. This means that pv ∈ C and pw = 0. In light of what has just been said and by (18), R satisfies

$$\displaystyle \begin{aligned} f(x_1,\ldots,x_n)(a-u)f(x_1,\ldots, x_n) \end{aligned} $$

(19)

that is a = u.

Remark 1

We would like to remark that any conclusion of the previous Lemma implies that F and G are generalized derivations of R. Hence, in view of Theorem 2, the statement of Lemma 6 can be written as follows: there exists a′∈ Q _r such that F(x) = xa′ and G(x) = a′x for all x ∈ R.

Proposition 2

Let R be a prime ring of characteristic different from 2, Q _r be its right Martindale quotient ring and C be its extended centroid. Suppose that F, G are inner b-generalized skew derivations of R respectively defined as follows:

$$\displaystyle \begin{aligned}F(x)=ax+bqxq^{-1}c,\qquad G(x)=ux+pvxv^{-1}w\end{aligned}$$

for all x ∈ R and suitable fixed a, b, c, u, p, w, q, v ∈ Q _r , with invertible elements q, v of Q _r . Let f(x ₁, …, x _n) be a non-central multilinear polynomial over C with n non-commuting variables. If

$$\displaystyle \begin{aligned} F(f(r_1,\ldots, r_n))f(r_1, \ldots, r_n)-f(r_1,\ldots,r_n)G(f(r_1,\ldots, r_n))=0 \end{aligned}$$

for all r ₁, …, r _n ∈ R, then one of the following statements holds:

• 1. bq ∈ Z(R) and pv ∈ C;

• 2. bq ∈ Z(R) and v ⁻¹ w ∈ C;

• 3. q ⁻¹ c ∈ Z(R) and pv ∈ C;

• 4. q ⁻¹ c ∈ Z(R) and v ⁻¹ w ∈ C.

In other words, F and G are generalized derivations of R and one of the following statements holds:

• 1. there exists a′∈ Q _r such that, F(x) = xa′ and G(x) = a′x for all x ∈ R;

• 2. f(x ₁, …, x _n)² is central-valued on R and there exist a′, b′∈ Q _r such that F(x) = a′x + xb′, G(x) = b′x + xa′, for all x ∈ R.

Proof

If R does not satisfy any non-trivial generalized polynomial identity, then the conclusion follows from Lemma 6. Therefore we may assume that

$$\displaystyle \begin{aligned} \begin{array}{ll} & \varPsi(x_1,\ldots,x_{n}) =\\ & \bigg(af(x_1,\ldots, x_n)+bqf(x_1,\ldots,x_n)q^{-1}c\bigg)f(x_1,\ldots,x_n)\\ & -f(x_1,\ldots,x_n)\bigg(uf(x_1,\ldots, x_n)+pvf(x_1, \ldots,x_n)v^{-1}w\bigg) \end{array} \end{aligned} $$

(20)

is a non-trivial generalized polynomial identity for R.

By Chuang [18] it follows that Ψ(x ₁, …, x _n) is a non-trivial generalized polynomial identity for Q _r. By the well-known Martindale’s theorem of [44], Q _r is a primitive ring having nonzero socle with the field C as its associated division ring. By Jacobson [31, Page 75] Q _r is isomorphic to a dense subring of the ring of linear transformations of a vector space V  over C, containing nonzero linear transformations of finite rank. Assume first that dim_C V = k ≥ 2 is a finite positive integer, then Q≅M _k(C) and the conclusion follows from Lemma 5.

Let us now consider the case of dim_C V = ∞. As in [48, Lemma 2], the set f(R) = {f(r ₁, …, r _n)|r _i ∈ R} is dense on R. By the fact that Ψ(r ₁, …, r _n) = 0 is a generalized polynomial identity of R, we know that R satisfies

$$\displaystyle \begin{aligned} \bigg(ax+bqxq^{-1}c\bigg)x-x\bigg(ux+pvxv^{-1}w\bigg). \end{aligned} $$

(21)

Recall that if an element r ∈ R centralizes the non-zero ideal H = soc(RC), then r ∈ C.

Hence we may assume there exist r ₁, r ₂, r ₃, r ₄ ∈ H = soc(RC) such that:

• 1. either [bq, r ₁] ≠ 0 or [pv, r ₁] ≠ 0;

• 2. either [bq, r ₂] ≠ 0 or [v ⁻¹ w, r ₂] ≠ 0

• 3. either [q ⁻¹ c, r ₃] ≠ 0 or [pv, r ₃] ≠ 0

• 4. either [q ⁻¹ c, r ₄] ≠ 0 or [v ⁻¹ w, r ₄] ≠ 0

and prove that a number of contradictions follows.

By Litoff’s Theorem [31, Page 90] there exists e ² = e ∈ H such that

• r ₁, r ₂, r ₃, r ₄ ∈ eRe;

• ar ₁, r ₁ a, ar ₂, r ₂ a, ar ₃, r ₃ a, ar ₄, r ₄ a ∈ eRe;

• br ₁, r ₁ b, br ₂, r ₂ b, br ₃, r ₃ b, br ₄, r ₄ b ∈ eRe;

• cr ₁, r ₁ c, cr ₂, r ₂ c, cr ₃, r ₃ c, cr ₄, r ₄ c ∈ eRe;

• qr ₁, r ₁ q, qr ₂, r ₂ q, qr ₃, r ₃ q, qr ₄, r ₄ q ∈ eRe;

• ur ₁, r ₁ u, ur ₂, r ₂ u, ur ₃, r ₃ u, ur ₄, r ₄ u ∈ eRe;

• pr ₁, r ₁ p, pr ₂, r ₂ p, pr ₃, r ₃ p, pr ₄, r ₄ p ∈ eRe;

• vr ₁, r ₁ v, vr ₂, r ₂ v, vr ₃, r ₃ v, vr ₄, r ₄ v ∈ eRe;

• wr ₁, r ₁ w, wr ₂, r ₂ w, wr ₃, r ₃ w, wr ₄, r ₄ w ∈ eRe;

• pvr ₁, r ₁ pv, pvr ₂, r ₂ pv, pvr ₃, r ₃ pv, pvr ₄, r ₄ pv ∈ eRe;

• bqr ₁, r ₁ bq, bqr ₂, r ₂ bq, bqr ₃, r ₃ bq, bqr ₄, r ₄ bq ∈ eRe;

• q ⁻¹ cr ₁, r ₁ q ⁻¹ c, q ⁻¹ cr ₂, r ₂ q ⁻¹ c, q ⁻¹ cr ₃, r ₃ q ⁻¹ c, q ⁻¹ cr ₄, r ₄ q ⁻¹ c ∈ eRe;

• v ⁻¹ wr ₁, r ₁ v ⁻¹ w, v ⁻¹ wr ₂, r ₂ v ⁻¹ w, v ⁻¹ wr ₃, r ₃ v ⁻¹ w, v ⁻¹ wr ₄, r ₄ v ⁻¹ w ∈ eRe,

where eRe≅M _m(C), the matrix ring over the extended centroid C. Note that eRe satisfies (21). By the above Lemma 5, we have that one of the following assertions holds:

• 1. ebqe ∈ C and epve ∈ C, which contradicts with the choice of r ₁ ∈ H;

• 2. ebqe ∈ C and ev ⁻¹ we ∈ C, which contradicts with the choice of r ₂ ∈ H;

• 3. eq ⁻¹ ce ∈ C and epve ∈ C, which contradicts with the choice of r ₃ ∈ H;

• 4. eq ⁻¹ ce ∈ C and ev ⁻¹ we ∈ C, which contradicts with the choice of r ₄ ∈ H.

As an easy consequence of Proposition 2 we also obtain a reduced version of Theorem 1 for the case both F and G are inner b-generalized derivations of R:

Proposition 3

Let R be a prime ring of characteristic different from 2, Q _r be its right Martindale quotient ring and C be its extended centroid. Suppose that F, G are inner b-generalized derivations of R respectively defined as follows:

$$\displaystyle \begin{aligned}F(x)=ax+bxc,\qquad G(x)=px+qxv\end{aligned}$$

for all x ∈ R and suitable fixed a, b, c, p, q, v ∈ Q _r . Let f(x ₁, …, x _n) be a non-central multilinear polynomial over C with n non-commuting variables. If

$$\displaystyle \begin{aligned}F(f(r_1,\ldots, r_n))f(r_1, \ldots, r_n)-f(r_1,\ldots,r_n)G(f(r_1,\ldots, r_n))=0\end{aligned}$$

for all r ₁, …, r _n ∈ R, then one of the following holds:

• 1. there exists a′∈ Q _r such that, F(x) = xa′ and G(x) = a′x for all x ∈ R;

• 2. f(x ₁, …, x _n)² is central-valued on R and there exist a′, b′∈ Q _r such that F(x) = a′x + xb′, G(x) = b′x + xa′, for all x ∈ R.

We are now ready to prove the more general result of this section.

We permit the following facts:

Note 15

Let R be a non-commutative prime ring, a, b ∈ R such that axb ∈ Z(R), for all x ∈ R. Then either a = 0 or b = 0.

Proof

We assume that a ≠ 0 and b ≠ 0. For any x ∈ R and by our assumption, both a(xb) ∈ Z(R) and a(xb)b ∈ Z(R). Thus we have that either b ∈ Z(R) or axb = 0 for all x ∈ R. In the first case it follows that aR ⊆ Z(R), which contradicts with the non-commutativity of R. In the latter case, by the primeness of R, we have the required conclusion.

Note 16

Let R be a non-commutative prime ring, a, b ∈ R, f(x ₁, …, x _n) a polynomial over C, which is not central valued on R. If af(r ₁, …, r _n)b ∈ Z(R), for all r ₁, …, r _n ∈ R, then either a = 0 or b = 0.

Proof

Let S be the additive subgroup of R generated by {f(y ₁, …, y _n) : y _i ∈ R}. Since f(y ₁, …, y _n) is not central and char(R) ≠ 2, it is well known that S contains a non-central Lie ideal L of R (see [17]). Moreover, since L is not central then there exists a non-central ideal I of R such that [I, R] ⊆ L. Therefore a[i, r]b ∈ Z(R), for any i ∈ I, r ∈ R. Since I and Q _r satisfy the same generalized identities it follows that a[x, y]b ∈ C for any x, y ∈ Q _r. In this situation we may apply the main result in [3] and one of the following holds: either a = 0 or b = 0 or Q _r is a central simple algebra of dimension at most 4 over C. Moreover, since Q _r is not commutative, then Q _r contains some non-trivial idempotent elements e = e ². In this last case, by the main hypothesis, one has a[e, x(1 − e)]b ∈ C, that is aex(1 − e)b ∈ C, for all x ∈ Q _r. By Note 15, either ae = 0 or (1 − e)b = 0.

If ae = 0 and by a[y, ex]b ∈ C, we get ayexb ∈ C, for any x, y ∈ Q _r. Thus, using Note 15 and since e ≠ 0, it follows that either a = 0 or b = 0, as required.

On the other hand, if (1 − e)b = 0 and by a[x, y(1 − e)]b ∈ C, we have that ay(1 − e)xb ∈ C, for any x, y ∈ Q _r. Once again by Note 15 and since e ≠ 1, we get their required conclusion.

Theorem 3

Let R be a prime ring of characteristic different from 2, Q _r be its right Martindale quotient ring and C be its extended centroid. Suppose that F, G are inner b-generalized skew derivations of R respectively defined as follows:

$$\displaystyle \begin{aligned}F(x)=ax+b\alpha(x)c,\qquad G(x)=ux+p\beta(x)w\end{aligned}$$

for all x ∈ R and suitable fixed a, b, c, u, p, w ∈ Q _r , and α, β ∈Aut(Q _r). Let f(x ₁, …, x _n) be a non-central multilinear polynomial over C with n non-commuting variables. If

$$\displaystyle \begin{aligned} F(f(r_1,\ldots, r_n))f(r_1, \ldots, r_n)-f(r_1,\ldots,r_n)G(f(r_1,\ldots, r_n))=0 \end{aligned} $$

(22)

for all r ₁, …, r _n ∈ R, then one of the following statements holds:

• 1. α = β = id, where id denotes the identical mapping on Q _r;

• 2. α = id and there exists an invertible element v ∈ Q _r such that β(x) = vxv ⁻¹ , for all x ∈ R;

• 3. β = id and there exists an invertible element q ∈ Q _r such that α(x) = qxq ⁻¹ , for all x ∈ R;

• 4. β = id and b = 0;

• 5. β = id and c = 0;

• 6. α = id and p = 0;

• 7. α = id and w = 0;

• 8. there exists an invertible element q ∈ Q _r such that α(x) = qxq ⁻¹ , for all x ∈ R, and either p = 0 or w = 0;

• 9. there exists an invertible element v ∈ Q _r such that β(x) = vxv ⁻¹ , for all x ∈ R, and either b = 0 or c = 0;

• 10. b = p = 0;

• 11. b = w = 0;

• 12. c = p = 0;

• 13. c = w = 0;

• 14. there exist invertible elements q, v ∈ Q _r such that α(x) = qxq ⁻¹ and β(x) = vxv ⁻¹ , for all x ∈ R.

In other words one of the following occurs:

• F and G are ordinary generalized derivations of R.

• F and G are inner b-generalized derivations;

• F and G are inner b-generalized skew derivations of R, associated with inner automorphisms;

In any case, respectively in light of Propositions 1, 3 and 2 , we have that one of the following statements holds:

• 1. there exists a′∈ Q _r such that, F(x) = xa′ and G(x) = a′x for all x ∈ R;

• 2. f(x ₁, …, x _n)² is central-valued on R and there exist a′, b′∈ Q _r such that F(x) = a′x + xb′, G(x) = b′x + xa′, for all x ∈ R.

Proof

On the contrary, we assume that the following hold simultaneously:

• either α ≠ id or β ≠ id;

• either α ≠ id or β is not an inner automorphism on Q _r;

• either β ≠ id or α is not an inner automorphism on Q _r;

• either α ≠ id or b ≠ 0;

• either α ≠ id or c ≠ 0;

• either β ≠ id or p ≠ 0;

• either β ≠ id or w ≠ 0;

• either α is not inner, or both p ≠ 0 and w ≠ 0;

• either β is not inner, or both b ≠ 0 and c ≠ 0;

• either b ≠ 0 or p ≠ 0;

• either b ≠ 0 or w ≠ 0;

• either c ≠ 0 or p ≠ 0;

• either c ≠ 0 or w ≠ 0;

• at least one among α and β is not an inner automorphism of R.

By our assumption R satisfies the following generalized polynomial

$$\displaystyle \begin{aligned} \begin{array}{ll} & \bigg(af(x_1,\ldots,x_n)+b\alpha\bigl(f(x_1,\ldots,x_n)\bigr)c\bigg)f(x_1,\ldots,x_n)\\ & -f(x_1,\ldots,x_n)\bigg(uf(x_1,\ldots,x_n)+p\beta\bigl(f(x_1,\ldots,x_n)\bigr)w\bigg). \end{array} \end{aligned} $$

(23)

In view of the Note 6, Q _r satisfies (23).

In case α = id, then β is not inner. Thus, by (23), Q _r satisfies the generalized polynomial

$$\displaystyle \begin{aligned} \begin{array}{ll} & \bigg(af(x_1,\ldots,x_n)+b(f(x_1,\ldots,x_n)c\bigg)f(x_1,\ldots,x_n)\\ & -f(x_1,\ldots,x_n)\bigg(uf(x_1,\ldots,x_n)+pf^\beta(y_1,\ldots,y_n)w\bigg). \end{array} \end{aligned} $$

(24)

In particular, pf ^β(y ₁, …, y _n)w is a generalized polynomial identity for Q _r. It is easy to see that pXw = 0, for any X ∈ S, the additive subgroup of Q _r generated by {f ^β(y ₁, …, y _n) : y _i ∈ Q _r}. Since f ^β(y ₁, …, y _n) is not central and char(Q _r) ≠ 2, it is well known that S must contain a non-central Lie ideal L. This implies pLw = (0) and, by the primeness of Q _r we get the contradiction that either p = 0 or w = 0.

Similarly, if we assume that β = id, then we obtain the contradiction that either b = 0 or c = 0.

Thus we may suppose both α ≠ id and β ≠ id. In what follows we denote f ^α(x ₁, …, x _n) = α(f(x ₁, …, x _n)).

If α and β are mutually outer, then by (23), Q _r satisfies

$$\displaystyle \begin{aligned} \begin{array}{ll} & \bigg(af(x_1,\ldots,x_n)+bf^\alpha(y_1,\ldots,y_n)c\bigg)f(x_1,\ldots,x_n)\\ & -f(x_1,\ldots,x_n)\bigg(uf(x_1,\ldots,x_n)+pf^\beta(z_1,\ldots,z_n)w\bigg). \end{array} \end{aligned} $$

(25)

In particular, Q _r satisfies both

$$\displaystyle \begin{aligned}bf^\alpha(y_1,\ldots,y_n)cf(x_1,\ldots,x_n)\end{aligned}$$

and

$$\displaystyle \begin{aligned}f(x_1,\ldots,x_n)pf^\beta(z_1,\ldots,z_n)w.\end{aligned}$$

Applying twice Corollary 2 to both last relations yields that either b = 0 or c = 0 and simultaneously either p = 0 or w = 0, which is a contradiction.

Assume finally that α and β are not mutually outer, then exists an invertible element q ∈ Q _r such that αβ ⁻¹(x) = qxq ⁻¹, for any x ∈ R. Therefore α(x) = qβ(x)q ⁻¹ and by (23) it follows that Q _r satisfies

$$\displaystyle \begin{aligned} \begin{array}{ll} & \bigg(af(x_1,\ldots,x_n)+bq\beta\bigl(f(x_1,\ldots,x_n)\bigr)q^{-1}c\bigg)f(x_1,\ldots,x_n)\\ & -f(x_1,\ldots,x_n)\bigg(uf(x_1,\ldots,x_n)+p\beta\bigl(f(x_1,\ldots,x_n)\bigr)w\bigg). \end{array} \end{aligned} $$

(26)

If β is an inner automorphism of Q _r, then the required conclusion follows from Proposition 2. On the other hand, if β is outer, then, by (26) we have that Q _r satisfies

$$\displaystyle \begin{aligned} \begin{array}{ll} & \bigg(af(x_1,\ldots,x_n)+bqf^\beta(y_1,\ldots,y_n)q^{-1}c\bigg)f(x_1,\ldots,x_n)\\ & -f(x_1,\ldots,x_n)\bigg(uf(x_1,\ldots,x_n)+pf^\beta(y_1,\ldots,y_n)w\bigg) \end{array} \end{aligned} $$

(27)

and in particular

$$\displaystyle \begin{aligned} bqf^\beta(y_1,\ldots,y_n)q^{-1}cf(x_1,\ldots,x_n)-f(x_1,\ldots,x_n)pf^\beta(y_1,\ldots,y_n)w \end{aligned} $$

(28)

is a generalized polynomial identity for Q _r. Since f(x ₁, …, x _n) is not central valued and in light of Lemma 3, one has that bqf ^β(y ₁, …, y _n)q ⁻¹ c = pf ^β(y ₁, …, y _n)w ∈ C for any y ₁, …, y _n ∈ Q _r. Hence Note 16 implies that the following hold simultaneously:

• either b = 0 or c = 0;

• either p = 0 or w = 0

and in any case we get a contradiction.

6 Commuting b-Generalized Derivations on Multilinear Polynomials

In this section we provide a proof of Theorem 1 in the case both F and G are arbitrary b-generalized derivations (not necessarily inner) and prove the following:

Theorem 4

Let R be a prime ring of characteristic different from 2, Q _r be its right Martindale quotient ring and C be its extended centroid, F and G non-zero b-generalized derivations of R. Suppose that f(x ₁, …, x _n) is a non-central multilinear polynomial over C such that F(f(X))f(X) − f(X)G(f(X)) = 0, for all X = (x ₁, …, x _n) ∈ R ⁿ , then one of the following statements holds:

• 1. there exists u ∈ Q _r such that, F(x) = xu and G(x) = ux for all x ∈ R;

• 2. f(x ₁, …, x _n)² is central-valued on R and there exist a, b ∈ Q _r such that F(x) = ax + xb, G(x) = bx + xa, for all x ∈ R.

Hence F and G are generalized derivations of R.

Proof

As mentioned in the Introduction, we can write F(x) = ax + bd(x), G(x) = px + qδ(x) for all x ∈ R, where a, b, p, q ∈ Q _r and d, δ are derivations of R. In light of Proposition 3, we may assume that:

• At least one among d and δ is not an inner derivation of R;

• At least one among b and q is not zero;

• If d is an inner derivation of R, then δ ≠ 0 and q ≠ 0;

• If δ is an inner derivation of R then d ≠ 0 and b ≠ 0.

We will prove that, under these assumptions, a number of contradiction follows.

Assume first that d and δ are both non-zero derivations and linearly C-independent modulo Q _r-inner derivations. Since Q _r satisfies

$$\displaystyle \begin{aligned} \begin{array}{ll} & \bigg( af(x_1,\ldots,x_n)+ bf^d(x_1,\ldots,x_n)+b\sum_{i=1}^n f(x_1,\ldots,d(x_i),\ldots,x_n)\bigg)f(x_1,\ldots,x_n)\\ & -f(x_1,\ldots,x_n)\bigg(pf(x_1,\ldots,x_n)+qf^\delta(x_1,\ldots,x_n)+q\sum_{i=1}^n f(x_1,\ldots,\delta(x_i),\ldots,x_n)\bigg) \end{array} \end{aligned} $$

(29)

and by Kharchenko [32], we arrive at that Q _r satisfies

$$\displaystyle \begin{aligned} \begin{array}{ll} & \bigg( af(x_1,\ldots,x_n)+ bf^d(x_1,\ldots,x_n)+b\sum_{i=1}^n f(x_1,\ldots,y_i,\ldots,x_n)\bigg)f(x_1,\ldots,x_n)\\ & -f(x_1,\ldots,x_n)\bigg(pf(x_1,\ldots,x_n)+qf^\delta(x_1,\ldots,x_n)+q\sum_{i=1}^n f(x_1,\ldots,z_i,\ldots,x_n)\bigg). \end{array} \end{aligned} $$

(30)

In particular, Q _r satisfies the blended components

$$\displaystyle \begin{aligned} bf(y_1,x_2,\ldots,x_n)\cdot f(x_1,\ldots,x_n) \end{aligned}$$

and

$$\displaystyle \begin{aligned} f(x_1,\ldots,x_n)\cdot q\cdot f(y_1,x_2,\ldots,x_n), \end{aligned}$$

which imply the contradiction b = 0 (by Corollary 2) and q = 0 (by Corollary 1).

Assume now that d and δ are both non-zero derivations and C-dependent modulo Q _r-inner derivations. Without loss of generality, we assume that δ = λd + ad _w, that is δ(x) = λd(x) + [w, x], for suitable 0 ≠ λ ∈ C and w ∈ Q _r. Moreover, in light of the previous remarks, d is not an inner derivation of R. By the hypothesis we have that

$$\displaystyle \begin{aligned} \begin{array}{ll} & \bigg( af(x_1,\ldots,x_n)+ bf^d(x_1,\ldots,x_n)+b\sum_{i=1}^n f(x_1,\ldots,d(x_i),\ldots,x_n)\bigg)f(x_1,\ldots,x_n)\\ & -f(x_1,\ldots,x_n)\bigg(pf(x_1,\ldots,x_n)+\lambda qf^d(x_1,\ldots,x_n)+\\ & + \lambda q\sum_{i=1}^n f(x_1,\ldots,d(x_i),\ldots,x_n)+ q[w,f(x_1,\ldots,x_n)]\bigg) \end{array} \end{aligned} $$

(31)

is a differential polynomial identity for Q _r, and again by Kharchenko [32] it follows that Q _r satisfies

$$\displaystyle \begin{aligned} \begin{array}{ll} & \bigg( af(x_1,\ldots,x_n)+ bf^d(x_1,\ldots,x_n)+b\sum_{i=1}^n f(x_1,\ldots,y_i,\ldots,x_n)\bigg)f(x_1,\ldots,x_n)\\ & - f(x_1,\ldots,x_n)\bigg(pf(x_1,\ldots,x_n)+\lambda qf^d(x_1,\ldots,x_n)+\\ & + \lambda q\sum_{i=1}^n f(x_1,\ldots,y_i,\ldots,x_n)+ q[w,f(x_1,\ldots,x_n)]\bigg). \end{array} \end{aligned} $$

(32)

In particular, Q _r satisfies the blended component

$$\displaystyle \begin{aligned} b\sum_if(x_1,\ldots,y_i,\ldots,x_n)f(x_1,\ldots,x_n)-\lambda f(x_1,\ldots,x_n)q \sum_if(x_1,\ldots,y_i,\ldots,x_n). \end{aligned} $$

(33)

Let us choose y ₂ = y ₃ = … = y _n = 0 and y ₁ = x ₁ in (33). This yields that Q _r satisfies

$$\displaystyle \begin{aligned} bf(x_1,\ldots,x_n)^2-\lambda f(x_1,\ldots,x_n)q f(x_1,\ldots,x_n). \end{aligned} $$

(34)

Moreover, for z∉C and y _i = [z, x _i] for any i = 1, …, n in (33), we also have that

$$\displaystyle \begin{aligned} b[z,f(x_1,\ldots,x_n)]f(x_1,\ldots,x_n)-\lambda f(x_1,\ldots,x_n)q [z,f(x_1,\ldots,x_n)] \end{aligned} $$

(35)

is a generalized polynomial identity for Q _r. Application of Proposition 1 to (34) implies that b = λq ∈ C. Therefore, by (35) it follows that Q _r satisfies

$$\displaystyle \begin{aligned}b[z,f(x_1,\ldots,x_n)]_2.\end{aligned}$$

Since z∉C and since neither char(R) = 2 nor f(x ₁, …, x _n) is central-valued on R, by Liu [38] we get b = 0, and so also q = 0, which is a contradiction.

We finally consider the case either d = 0 or δ = 0. Without loss of generality, we may assume δ = 0 (the case d = 0 is similar and we omit it for brevity). By our assumption it follows that Q _r satisfies

$$\displaystyle \begin{aligned} \begin{array}{ll} & \bigg( af(x_1,\ldots,x_n)+ bf^d(x_1,\ldots,x_n)+b\sum_{i=1}^n f(x_1,\ldots,d(x_i),\ldots,x_n)\bigg)f(x_1,\ldots,x_n)\\ & - f(x_1,\ldots,x_n)pf(x_1,\ldots,x_n). \end{array} \end{aligned} $$

(36)

Moreover, as above remarked, in this case d is not an inner derivation of R. In view of Kharchenko’s theorem in [32], Q _r satisfies

$$\displaystyle \begin{aligned} \begin{array}{ll} & \bigg( af(x_1,\ldots,x_n)+ bf^d(x_1,\ldots,x_n)+b\sum_{i=1}^n f(x_1,\ldots,y_i,\ldots,x_n)\bigg)f(x_1,\ldots,x_n)\\ & - f(x_1,\ldots,x_n)pf(x_1,\ldots,x_n). \end{array} \end{aligned} $$

(37)

Therefore

$$\displaystyle \begin{aligned} bf(y_1,x_2,\ldots,x_n)f(x_1,\ldots,x_n) \end{aligned} $$

(38)

is a generalized polynomial identity for Q _r, implying again the contradiction b = 0.

7 The Main Result

The last part of our paper is dedicated to the proof of Theorem 1 in its most general form. For sake of clearness and completeness, we recall our hypothesis.

We assume that R is a prime ring of characteristic different from 2, Q _r its right Martindale quotient ring and C its extended centroid, α ∈Aut(R), d and δ skew derivations of R associated with α, such that both d and δ are commuting with α. We suppose that F, G are b-generalized skew derivations of R, respectively associated with terms (b, α, d) and (p, β, δ). We may write F(x) = ax + bd(x) and G(x) = ux + pδ(x), for all x ∈ R and suitable a, u ∈ Q _r. We assume that f(x ₁, …, x _n) is a non-central multilinear polynomial over C with n non-commuting variables, such that

$$\displaystyle \begin{aligned} F(f(r_1,\ldots, r_n))f(r_1, \ldots, r_n)-f(r_1,\ldots,r_n)G(f(r_1,\ldots, r_n))=0\end{aligned} $$

(39)

for all r ₁, …, r _n ∈ R, that is R satisfies

$$\displaystyle \begin{aligned} \begin{array}{ll} & \bigg(af(x_1,\ldots,x_n)+bd(f(x_1,\ldots,x_n))\bigg)f(x_1,\ldots,x_n)\\ & -f(x_1,\ldots,x_n)\bigg(uf(x_1,\ldots,x_n)+p\delta(f(x_1,\ldots,x_n))\bigg). \end{array} \end{aligned} $$

(40)

Under these assumptions, we’ll prove that one of the following statements holds:

• 1. d = δ = 0;

• 2. α = id;

• 3. there exist b′, c′∈ Q _r such that d(x) = b′x − α(x)b′ and δ(x) = c′x − α(x)c′, for all x ∈ R;

• 4. b = p = 0;

• 5. b = 0 and δ = 0;

• 6. p = 0 and d = 0.

In other words, either F and G are generalized derivations of R, or F and G are b-generalized derivations of R, or F and G are inner b-generalized skew derivations of R. Therefore, respectively in light of Theorems 2, 4 and 3, we have that one of the following holds:

• 1. there exists a′∈ Q _r such that, F(x) = xa′ and G(x) = a′x for all x ∈ R;

• 2. f(x ₁, …, x _n)² is central valued on R and there exist a′, b′∈ Q _r such that F(x) = a′x + xb′, G(x) = b′x + xa′, for all x ∈ R.

Proof of Theorem 1

By (40) and Note 9,

$$\displaystyle \begin{aligned} \begin{array}{ll} & \bigg(af(x_1,\ldots,x_n)+bf^d(x_1,\ldots,x_n)+\\ & b\sum_{\sigma \in S_n} \alpha(\gamma_\sigma)\sum_{j=0}^{n-1}\alpha(x_{\sigma(1)}\cdot x_{\sigma(2)}\cdots x_{\sigma(j)})d(x_{\sigma(j+1)})x_{\sigma(j+2)}\cdots x_{\sigma(n)}\bigg)f(x_1,\ldots,x_n)\\ & -f(x_1,\ldots,x_n)\bigg(uf(x_1,\ldots,x_n)+pf^\delta(x_1,\ldots,x_n)\\ & +p\sum_{\sigma \in S_n} \beta(\gamma_\sigma)\sum_{j=0}^{n-1}\alpha(x_{\sigma(1)}\cdot x_{\sigma(2)}\cdots x_{\sigma(j)})\delta(x_{\sigma(j+1)})x_{\sigma(j+2)}\cdots x_{\sigma(n)}\bigg) \end{array} \end{aligned} $$

(41)

is a generalized identity for R.

On the contrary we assume that the following hold simultaneously:

• either d ≠ 0 or δ ≠ 0;

• α ≠ id;

• at least one among d and δ is not an inner skew derivation of R;

• at least one among b and p is not zero;

• at least one among b or δ is not zero;

• at least one among p or d is not zero.

7.1 Let d and δ be C −Linearly Independent Modulo SD_int

In this case, in view of (41) we know that R satisfies the generalized polynomial

$$\displaystyle \begin{aligned} \begin{array}{ll} & \bigg(af(x_1,\ldots,x_n)+bf^d(x_1,\ldots,x_n)+\\ & b\sum_{\sigma \in S_n} \alpha(\gamma_\sigma)\sum_{j=0}^{n-1}\alpha(x_{\sigma(1)}\cdot x_{\sigma(2)}\cdots x_{\sigma(j)})y_{\sigma(j+1)}x_{\sigma(j+2)}\cdots x_{\sigma(n)}\bigg)f(x_1,\ldots,x_n)\\ & -f(x_1,\ldots,x_n)\bigg(uf(x_1,\ldots,x_n)+pf^\delta(x_1,\ldots,x_n)\\ & +p\sum_{\sigma \in S_n} \beta(\gamma_\sigma)\sum_{j=0}^{n-1}\alpha(x_{\sigma(1)}\cdot x_{\sigma(2)}\cdots x_{\sigma(j)})z_{\sigma(j+1)}x_{\sigma(j+2)}\cdots x_{\sigma(n)}\bigg). \end{array} \end{aligned} $$

(42)

In particular, R satisfies any blended component

$$\displaystyle \begin{aligned} b\bigg(\sum_{\sigma \in S_n} \alpha(\gamma_\sigma)\sum_{i=1}^{n}\alpha(x_{\sigma(1)}\cdot x_{\sigma(2)}\cdots x_{\sigma(i-1)})y_{\sigma(i)}x_{\sigma(i+1)}\cdots x_{\sigma(n)}\bigg)f(x_1,\ldots,x_n).\end{aligned} $$

(43)

In light of the Note 6, Q _r satisfies (43).

Suppose there exists an invertible element q ∈ Q _r such that α(x) = qxq ⁻¹ for all x ∈ Q _r. Since α ≠ id ∈Aut(R), we may assume q∉C. Moreover, it is clear that α(γ _σ) = γ _σ for all coefficients involved in f(x ₁, …, x _n). If we replace each y _σ(i) with qx _σ(i) in (43), then Q _r satisfies the generalized polynomial

$$\displaystyle \begin{aligned} b\bigg(q\sum_{\sigma \in S_n}\gamma_\sigma x_{\sigma(1)}\cdot x_{\sigma(2)}\cdots x_{\sigma(i-1)}x_{\sigma(i)}x_{\sigma(i+1)}\cdots x_{\sigma(n)}\bigg) f(x_1,\ldots,x_n).\end{aligned} $$

That is bqf(x ₁, …, x _n)² = 0, which implies bq = 0. Since q is invertible, we obtain that b = 0.

Finally, assume that α is outer. By (43) it follows that Q _r satisfies the generalized polynomial

$$\displaystyle \begin{aligned} b\bigg(\sum_{\sigma \in S_n} \alpha(\gamma_\sigma)\sum_{i=1}^{n}z_{\sigma(1)}\cdot z_{\sigma(2)}\cdots z_{\sigma(i-1)}y_{\sigma(i)}x_{\sigma(i+1)}\cdots x_{\sigma(n)}\bigg)f(x_1,\ldots,x_n). \end{aligned} $$

(44)

For any i = 1, …, n, Q _r also satisfies the generalized polynomial

$$\displaystyle \begin{aligned} b\bigg(\sum_{\sigma \in S_{n-1}} \alpha(\gamma_\sigma)z_{\sigma(1)}\cdot z_{\sigma(2)}\cdots z_{\sigma(i-1)}\cdot z_{\sigma(i+1)}\cdots z_{\sigma(n)}\cdot y_{i}\bigg)f(x_1,\ldots,x_n). \end{aligned} $$

(45)

Let us write

$$\displaystyle \begin{aligned} \sum_{\sigma \in S_{n-1}} \alpha(\gamma_\sigma) x_{\sigma(1)}\cdots x_{\sigma(j-i)} x_{\sigma(j+1)}\cdots x_{\sigma(n)}=t_j(x_1,\ldots,x_{j-1},x_{j+1},\ldots,x_n), \end{aligned}$$

where any t _j is a multilinear polynomial of degree n − 1 and x _j never appears in any monomial of t _j. It follows from (45) that Q _r satisfies the generalized polynomial

$$\displaystyle \begin{aligned} bt_j(z_1,\ldots,z_{j-1},z_{j+1},\ldots,z_n)f(x_1,\ldots,x_n). \end{aligned}$$

As a consequence of Lemma 3 and Corollary 2, either b = 0 or t _j(z ₁, …, z _j−1, z _j+1, …, z _n) is a generalized polynomial identity for Q _r for all j = 1, …, n. Moreover, we also denote f ^α(x ₁, …, x _n) the polynomial obtained from f(x ₁, …, x _n) by replacing each coefficient γ _σ with α(γ _σ) and notice that f ^α(r ₁, …, r _n) ≠ 0. Hence, in the case t _j(z ₁, …, z _j−1, z _j+1, …, z _n) is a generalized polynomial identity for Q _r for all j = 1, …, n, and since

$$\displaystyle \begin{aligned} f^\alpha(x_1,\ldots,x_n)=\sum_j x_{j}t_j(x_1,\ldots,x_{j-1},x_{j+1},\ldots,x_n), \end{aligned}$$

f ^α(x ₁, …, x _n) is a generalized polynomial identity for Q _r, which is also a contradiction. Thus we conclude again that b = 0.

The previous argument shows that b = 0 in any case.

Moreover, by (42) it follows that Q _r satisfies

$$\displaystyle \begin{aligned} f(x_1,\ldots,x_n)p\bigg(\sum_{\sigma \in S_n} \beta(\gamma_\sigma)\sum_{i=1}^{n}\alpha(x_{\sigma(1)}\cdot x_{\sigma(2)}\cdots x_{\sigma(i-1)})z_{\sigma(i)}x_{\sigma(i+1)}\cdots x_{\sigma(n)}\bigg). \end{aligned}$$

By using the same above argument, one can show that p = 0, which is a contradiction. We omit the proof for brevity.

7.2 Let d and δ be C −Linearly Dependent Modulo SD_int

We firstly assume that there exist 0 ≠ λ ∈ C, 0 ≠ μ ∈ C, c ∈ Q _r and γ ∈Aut(R) such that λd(x) + μδ(x) = cx − γ(x)c for all x ∈ R. Denote η = −μ ⁻¹ λ and q = μ ⁻¹ c. Thus δ(x) = ηd(x) + qx − γ(x)q for all x ∈ R. Therefore by (40), Q _r satisfies the generalized polynomial

$$\displaystyle \begin{aligned} \begin{array}{ll} & af(x_1,\ldots,x_n)^2+bd\bigl(f(x_1,\ldots,x_n)\bigr)f(x_1,\ldots,x_n)\\ & -f(x_1,\ldots,x_n)(u+pq)f(x_1,\ldots,x_n)-\eta f(x_1,\ldots,x_n)pd\bigl(f(x_1,\ldots,x_n)\bigr)\\ & +f(x_1,\ldots,x_n)p\gamma\bigl(f(x_1,\ldots,x_n)\bigr)q. \end{array} \end{aligned} $$

(46)

That is, Q _r satisfies the generalized polynomial

$$\displaystyle \begin{aligned} \begin{array}{ll} & af(x_1,\ldots,x_n)^2+bf^d(x_1,\ldots,x_n)f(x_1,\ldots,x_n)\\ & +b\bigg(\sum_{\sigma \in S_n} \alpha(\gamma_\sigma)\sum_{j=0}^{n-1}\alpha(x_{\sigma(1)}\cdot x_{\sigma(2)}\cdots x_{\sigma(j)})d(x_{\sigma(j+1)})x_{\sigma(j+2)}\cdots x_{\sigma(n)}\bigg)f(x_1,\ldots,x_n)\\ & -f(x_1,\ldots,x_n)(u+pq)f(x_1,\ldots,x_n)-\eta f(x_1,\ldots,x_n)pf^d(x_1,\ldots,x_n)\\ & -\eta f(x_1,\ldots,x_n)p\bigg(\sum_{\sigma \in S_n} \alpha(\gamma_\sigma)\sum_{j=0}^{n-1}\alpha(x_{\sigma(1)}\cdot x_{\sigma(2)}\cdots x_{\sigma(j)})d(x_{\sigma(j+1)})x_{\sigma(j+2)}\cdots x_{\sigma(n)}\bigg)\\ & +f(x_1,\ldots,x_n)p\gamma\bigl(f(x_1,\ldots,x_n)\bigr)q. \end{array} \end{aligned} $$

(47)

In case d is outer, by (47) Q _r satisfies the generalized polynomial

$$\displaystyle \begin{aligned} \begin{array}{ll} & af(x_1,\ldots,x_n)^2+bf^d(x_1,\ldots,x_n)f(x_1,\ldots,x_n)\\ & +b\bigg(\sum_{\sigma \in S_n} \alpha(\gamma_\sigma)\sum_{j=0}^{n-1}\alpha(x_{\sigma(1)}\cdot x_{\sigma(2)}\cdots x_{\sigma(j)})y_{\sigma(j+1)}x_{\sigma(j+2)}\cdots x_{\sigma(n)}\bigg)f(x_1,\ldots,x_n)\\ & -f(x_1,\ldots,x_n)(u+pq)f(x_1,\ldots,x_n)-\eta f(x_1,\ldots,x_n)pf^d(x_1,\ldots,x_n)\\ & -\eta f(x_1,\ldots,x_n)p\bigg(\sum_{\sigma \in S_n} \alpha(\gamma_\sigma)\sum_{j=0}^{n-1}\alpha(x_{\sigma(1)}\cdot x_{\sigma(2)}\cdots x_{\sigma(j)})y_{\sigma(j+1)}x_{\sigma(j+2)}\cdots x_{\sigma(n)}\bigg)\\ & +f(x_1,\ldots,x_n)p\gamma\bigl(f(x_1,\ldots,x_n)\bigr)q. \end{array} \end{aligned} $$

(48)

In particular,

$$\displaystyle \begin{aligned} \begin{array}{ll} & b\bigg(\sum_{\sigma \in S_n} \alpha(\gamma_\sigma)\sum_{j=0}^{n-1}\alpha(x_{\sigma(1)}\cdots x_{\sigma(j)})y_{\sigma(j+1)}x_{\sigma(j+2)}\cdots x_{\sigma(n)}\bigg)f(x_1,\ldots,x_n)\\ & -\eta f(x_1,\ldots,x_n)p\bigg(\sum_{\sigma \in S_n} \alpha(\gamma_\sigma)\sum_{j=0}^{n-1}\alpha(x_{\sigma(1)}\cdots x_{\sigma(j)})y_{\sigma(j+1)}x_{\sigma(j+2)}\cdots x_{\sigma(n)}\bigg) \end{array} \end{aligned} $$

(49)

is satisfied by R as well as Q _r (see Note 6 again).

Suppose there exists an invertible element w ∈ Q _r such that α(x) = wxw ⁻¹ for all x ∈ Q _r. Since α ≠ 1 ∈Aut(R), we may assume w∉C. As above, we remark that α(γ _σ) = γ _σ for all coefficients involved in f(x ₁, …, x _n). Therefore, if we replace each y _σ(i) with wx _σ(i) in (49), we obtain that Q _r satisfies the generalized polynomial

$$\displaystyle \begin{aligned} \bigg(bwf(x_1,\ldots,x_n)-f(x_1,\ldots,x_n)(\eta pw)\bigg)f(x_1,\ldots,x_n).\end{aligned}$$

Applying again Corollary 1 yields bw = ηpw ∈ C. In particular b = ηp. Let us now replace each y _σ(i) with w[z, x _σ(i)] in (49), for some element z∉C. Thus we obtain that Q _r satisfies the generalized polynomial

$$\displaystyle \begin{aligned} bw\bigg[z,f(x_1,\ldots,x_n)\bigg]_2. \end{aligned}$$

Since f(x ₁, …, x _n) is not central-valued on Q _r and z∉C, we get the contradiction b = p = 0.

Finally, assume that α is outer. By (49) we know that Q _r satisfies the generalized polynomial

$$\displaystyle \begin{aligned} \begin{array}{ll} & b \bigg(\sum_{\sigma \in S_n} \alpha(\gamma_\sigma)\sum_{i=1}^{n}z_{\sigma(1)}\cdot z_{\sigma(2)}\cdots z_{\sigma(i-1)}y_{\sigma(i)}x_{\sigma(i+1)}\cdots x_{\sigma(n)}\bigg)f(x_1,\ldots,x_n)\\ & -\eta f(x_1,\ldots, x_n)p\bigg(\sum_{\sigma \in S_n} \alpha(\gamma_\sigma)\sum_{i=1}^{n}z_{\sigma(1)}\cdot z_{\sigma(2)}\cdots z_{\sigma(i-1)}y_{\sigma(i)}x_{\sigma(i+1)}\cdots x_{\sigma(n)}\bigg)f(x_1,\ldots,x_n) \end{array} \end{aligned} $$

(50)

and, for any i = 1, …, n, Q _r also satisfies the generalized polynomial

$$\displaystyle \begin{aligned} \begin{array}{ll} & b\bigg(\sum_{\sigma \in S_{n-1}} \alpha(\gamma_\sigma)z_{\sigma(1)}\cdot z_{\sigma(2)}\cdots z_{\sigma(i-1)}\cdot z_{\sigma(i+1)}\cdots z_{\sigma(n)}\cdot y_{i}\bigg)f(x_1,\ldots,x_n)\\ & -\eta f(x_1,\ldots,x_n)p\bigg(\sum_{\sigma \in S_{n-1}} \alpha(\gamma_\sigma)z_{\sigma(1)}\cdot z_{\sigma(2)}\cdots z_{\sigma(i-1)}\cdot z_{\sigma(i+1)}\cdots z_{\sigma(n)}\cdot y_{i}\bigg). \end{array} \end{aligned} $$

(51)

As above, let us write

$$\displaystyle \begin{aligned} \sum_{\sigma \in S_{n-1}} \alpha(\gamma_\sigma) x_{\sigma(1)}\cdots x_{\sigma(j-i)} x_{\sigma(j+1)}\cdots x_{\sigma(n)}=t_j(x_1,\ldots,x_{j-1},x_{j+1},\ldots,x_n), \end{aligned}$$

where any t _j is a multilinear polynomial of degree n − 1 and x _j never appears in any monomial of t _j. In view of (51), we get

$$\displaystyle \begin{aligned} \begin{array}{ll} & b\bigg(t_j(z_1,\ldots,z_{j-1},z_{j+1},\ldots,z_n)y\bigg)f(x_1,\ldots,x_n)\\ & -\eta f(x_1,\ldots,x_n)p\bigg(t_j(z_1,\ldots,z_{j-1},z_{j+1},\ldots,z_n)y\bigg). \end{array} \end{aligned} $$

(52)

From Lemma 3 it follows that

$$\displaystyle \begin{aligned} bt_j(z_1,\ldots,z_{j-1},z_{j+1},\ldots,z_n)y=\eta pt_j(z_1,\ldots,z_{j-1},z_{j+1},\ldots,z_n)y \in C.\end{aligned} $$

(53)

Suppose that t _j(z ₁, …, z _j−1, z _j+1, …, z _n) is central-valued on Q _r for all j = 1, …, n. Since

$$\displaystyle \begin{aligned} f^\alpha(x_1,\ldots,x_n)=\sum_j x_{j}t_j(x_1,\ldots,x_{j-1},x_{j+1},\ldots,x_n), \end{aligned}$$

it follows that f ^α(x ₁, …, x _n) is a central-valued on Q _r, a contradiction. Therefore (53) forces b = 0 and ηp = 0, which is again a contradiction.

Let us next start from (46) and consider the case when d(x) = vx − α(x)v for all x ∈ R and for some fixed v ∈ Q _r. Hence, δ(x) = (ηv + q)x − α(x)ηv − γ(x)q. Therefore, by Note 8, F and G are simultaneously inner b-generalized skew derivations of R and, by Theorem 3 a number of contradictions follows.

We analyze now the last case. Let us start again from relation (40) and assume again that d and δ are C −linearly dependent modulo SD_int. That is λd(x) + μδ(x) = cx − γ(x)c for all x ∈ R. Moreover, in view of the previous argument, we have to assume now λ = 0. Thus δ(x) = qx − γ(x)q for all x ∈ R and q = μ ⁻¹ c. Therefore by (40), Q _r satisfies the generalized polynomial

$$\displaystyle \begin{aligned} \begin{array}{ll} & af(x_1,\ldots,x_n)^2+bd\bigl(f(x_1,\ldots,x_n)\bigr)f(x_1,\ldots,x_n)\\ & -f(x_1,\ldots,x_n)(u+pq)f(x_1,\ldots,x_n)\\ & +f(x_1,\ldots,x_n)p\gamma\bigl(f(x_1,\ldots,x_n)\bigr)q. \end{array} \end{aligned} $$

(54)

We finally observe that (54) is equivalent to (46) in case η = 0. Therefore the same above argument completes our proof.

8 Some Open Problems

In the light of the motivation and contents of this article, we will propose several topics for future research in this field. More precisely, some informations about the structure of a prime ring R and the description of all possible forms of a b-generalized skew derivation F of R can be obtained if one of the following conditions is satisfied:

• 1. F(x)ⁿ = 0 for all x ∈ L, where n is a fixed positive integer and L is a noncommutative Lie ideal of R.

• 2. F(x)ⁿ ∈ Z(R) for all x ∈ L, where n is a fixed positive integer and L is a noncommutative Lie ideal of R.

• 3. F(x)ⁿ ∈ Z(R) for all x ∈ I, where n is a fixed positive integer and I is a non-zero one sided ideal of R.

• 4. aF(x)ⁿ = 0 for all x ∈ I, where n is a fixed positive integer, I is an ideal of R and a is a non-zero element of R.