1 Introduction

Throughout this paper, Z(R) will denote the center of an associative ring R. The symbols \(x\circ y\) and [x, y], where x, \(y \in R\), stand for the Jordan product \(xy+yx\) and Lie product \(xy-yx\) respectively. An additive subgroup U of R is said to be a Lie ideal if \([U, R]\subseteq U\). A Lie ideal U of R is called square closed if \(u^2\in U\) for all \(u\in U\). For any a, \(b \in R\), a ring R is said to be prime if whenever \(aRb=0\) implies \(a=0\) or \(b=0\) and is semiprime if for any \(a \in R\), \(aRa=0\) implies \(a=0\). A mapping f is called an additive mapping on R if \(f(x+y)=f(x)+f(y)\) holds for all x, \(y \in R\). Let a mapping \(d: R\rightarrow R\) defined as \(d(xy)=d(x)y+xd(y)\) for all \(x, y \in R\). If d is an additive mapping, then d is said to be a derivation on R. Recall that an additive mapping f on R is said to be left multiplier if \(f(xy)=f(x)y\) for all \(x, y \in R\). An additive mapping \(F:R\rightarrow R\) is said to be a generalized derivation if there exists a derivation \(d:R \rightarrow R\) such that \(F(xy)=F(x)y+xd(y)\) for all \(x, y \in R\). Generalized derivations first time introduced by Bre\({\check{s}}\)ar in [7]. Obviously, every derivation is a generalized derivation but the converse need not be true in general. Hence generalized derivation covers both the concepts of derivation and left multiplier maps.

An additive mapping d on R is said to be a \((\alpha , \alpha )\)-derivation on R if \(d(xy)=d(x)\alpha (y)+\alpha (x)d(y)\) for all \(x, y\in R\). We notice that every (1, 1)-derivation is a ordinary derivation, where 1 denotes an identity mapping on R. An additive mapping F on R is called a generalized \((\alpha , \alpha )\)-derivation on R if there exists a \((\alpha , \alpha )\)-derivation d on R such that \(F(xy)=F(x)\alpha (y)+\alpha (x)d(y)\) for all \(x, y\in R\). Hence, every generalized \((\alpha , \alpha )\)-derivations covers the concepts of derivations as well as generalized derivations.

Let S be a non empty-subset of R. A mapping \(h:R\rightarrow R\) is called centralizing on S if \([h(x), x] \in Z(R)\) for all \(x \in S\) and is called commuting on S if \([h(x), x]=0\) for all \(x \in S\). In this direction, Posner in [11] was first who investigated the commutativity of ring. More precisely, He proved that: If R is a prime ring with nonzero derivation d on R such that d centralizing on R, then R is commutative. Further, regarding commutativity in prime ring, Ashraf and Rehman in [3], studied as follows: Let R be a prime ring and I a non-zero ideal of R. Suppose that d a non-zero derivation on R. If one of the following hold: (i) \(d(xy)+ xy \in Z(R)\); (ii) \(d(xy)-xy \in Z(R)\) for all \(x, y \in I\), then R must be commutative.

Further, Ashraf et al. in [4] extended their work, replacing derivation d with a generalized derivation F in a prime ring R. More precisely, they proved that:

Let R be a prime ring R and I a non-zero ideal of R. Suppose F is a generalized derivation associated with a non zero derivation d on R. If one of the following hold:

(i) \(F(xy)\pm xy \in Z(R)\); (ii) \(F(xy)\pm yx \in Z(R)\);

(iii) \(F(x)F(y)\pm xy \in Z(R)\)

for all \(x, y \in I\), then R is commutative.

Further, in [8] Dhara has studied the situations, when a generalized derivation F of a semiprime ring R acts as homomorphism or anti-homomorphism in a non-zero left ideal of R. Recently, Albas [1] studied the above mentioned identities in prime rings with central valued. Albas proved the following theorem:

Let R be a prime ring with center Z(R) and I be a non-zero ideal of R. If R admits a non-zero generalized derivation F of R, with associated derivation d such that \(F(xy)-F(x)F(y) \in Z(R)\) or \(F(xy)+F(x)F(y) \in Z(R)\) for all x, \(y \in I\), then either R is commutative or \(F=I_{id}\) or \(F=-I_{id}\), where \(I_{id}\) denotes the identity map of the ring R.

In several papers all these identities are also investigated in some appropriate subsets of prime and semiprime rings. In this view we refer to [1, 2, 5, 8, 9, 12, 13]; where further references can be found. Further Atteya in [5] continued these results on semiprime ring, stated as: Let R be a semiprime ring and I be a non-zero ideal of R. Then R contains a non-zero central ideal if one of the following condition hold; (i) \(F(xy) \pm xy \in Z(R)\), (ii) \(F(xy) \pm yx \in Z(R)\), (iii) \(F(x)F(y) \pm xy \in Z(R)\) for all \(x, y \in I\), where F is a generalized derivation associated a non-zero derivation d on R.

In this line of investigation, in the present paper, our aim is to investigate some of the above identities involving generalized \((\alpha , \alpha )\)-derivations on some suitable subsets in prime rings.

2 Preliminaries

We need the following lemmas to prove our Theorems.

Lemma 1

[6, Lemma 4] If \(U \not \subseteq Z(R)\) is a Lie ideal of a 2-torsion free prime ring R and \(a, b\in R\) such that \(aUb=0\), then either \(a=0\) or \(b=0\).

Lemma 2

[10, Lemma 2.5] Let R be a prime ring with char \((R)\ne 2\), and let U be a nonzero square-closed Lie ideal of R. If \([u, v]_{\alpha , \beta }=0\) for all \(u, v\in U\), then \(U\subseteq Z(R)\).

Lemma 3

[10, Lemma 2.6] Let R be a 2-torsion free prime ring and U a nonzero square-closed Lie ideal of R. Suppose that there exists a nonzero \((\alpha , \beta )\)-derivation d such that \(d(u)=0\) for all \(u\in U\). Then \(U\subseteq Z(R)\).

3 Results

Theorem 1

Let R be a 2-torsion free prime ring, U a nonzero square-closed Lie ideal of R. Suppose that G and F are two generalized \((\alpha , \alpha )\)-derivations associated with \((\alpha , \alpha )\)-derivations g and d respectively on R. If \(G(uv)\pm F(u)F(v)\in Z(R)\) for all \(u, v \in U\), then either \(d=g=0\) or \(U\subseteq Z(R)\).

Proof

If \(U \subseteq Z(R),\) then we are done. Now we assume that \(U \not \subseteq Z(R).\) By the hypothesis

$$\begin{aligned} G(uv)+F(u)F(v)\in Z(R) \end{aligned}$$
(3.1)

for all \(u, v \in U\). Replacing 2vw in place of v, we obtain

$$\begin{aligned} 2(G(uv)\alpha (w)+\alpha (uv)g(w)+F(u)F(v)\alpha (w)+F(u)\alpha (v)d(w)) \in Z(R) \end{aligned}$$
(3.2)

for all \(u, v, w \in U\). Since R is 2-torsion free, we obtain

$$\begin{aligned} (G(uv)+F(u)F(v))\alpha (w)+\alpha (uv)g(w)+F(u)\alpha (v)d(w) \in Z(R) \end{aligned}$$
(3.3)

for all \(u, v, w \in U\). Commuting (3.3) with \(\alpha (w)\), and using \(G(uv)+F(u)F(v)\in Z(R)\), we obtain

$$\begin{aligned}{}[\alpha (uv)g(w)+F(u)\alpha (v)d(w), \alpha (w)]=0 \end{aligned}$$
(3.4)

for all \(u, v, w \in U\). Substituting 2ux instead of u, where \(x\in U\), in (3.4), we obtain

$$\begin{aligned}{}[\alpha (uxv)g(w)+F(u)\alpha (xv)d(w)+\alpha (u)d(x)\alpha (v)d(w), \alpha (w)]=0 \end{aligned}$$
(3.5)

for all \(u, v, w, x \in U\). Replacing v by 2xv in (3.4) and using 2-torsion freeness of R, we get

$$\begin{aligned}{}[\alpha (uxv)g(w)+F(u)\alpha (xv)d(w), \alpha (w)]=0 \end{aligned}$$
(3.6)

for all \(u, v, w, x \in U\). Subtracting (3.6) from (3.5), we get

$$\begin{aligned}{}[\alpha (u)d(x)\alpha (v)d(w), \alpha (w)]=0 \end{aligned}$$
(3.7)

for all \(u, v, w, x \in U\). Again replacing u by 2yu in (3.7), where \(y\in U\), we obtain

$$\begin{aligned}{}[y, w]u\alpha ^{-1}(d(x))v\alpha ^{-1}(d(w))=0 \end{aligned}$$
(3.8)

for all \(u, v, w, x, y \in U\). In particular for \(x=w\), we get

$$\begin{aligned}{}[y, w]u\alpha ^{-1}(d(w))v\alpha ^{-1}(d(w))=0. \end{aligned}$$

By Lemma 1, it implies that either \(\alpha ^{-1}(d(w))=0\) or \([y, w]u\alpha ^{-1}(d(w))=0\) for all \(u, w, y \in U\). In any cases it follows that \([y, w]u\alpha ^{-1}(d(w))=0\) for all \(u, w, y \in U\). Again using Lemma 1, it gives that either \([y, w]=0\) or \(\alpha ^{-1}(d(w))=0\) for all \(w, y \in U\).

Suppose that \(U_1=\{w \in U \mid [y, w]=0\}\) and \(U_2=\{w \in U \mid \alpha ^{-1}(d(w))=0\}\). Clearly \(U_1\) and \(U_2\) are additive subgroups of U and \(U=U_1 \cup U_2\), so either \(U=U_1\) or \(U=U_2\). If \(U=U_1\), then by Lemma 2, \(U\subseteq Z(R)\), a contradiction. If \(U=U_2\), then \(\alpha ^{-1}(d(w))=0\) for all \(w\in U\). It implies that \(d(w)=0\) for all \(w\in U\). By Lemma 3, it gives \(d=0\).

If \(d=0\), then (3.6) reduces to \([\alpha (uxv)g(w), \alpha (w)]=0\) for all \(u, x, v, w\in U\). Replacing u by 2yu, we obtain \([\alpha (y), \alpha (w)]\alpha (uxv)g(w)=0\). It implies that

$$\begin{aligned}{}[y, w]uxv\alpha ^{-1}(g(w))=0 \end{aligned}$$

for all \(u, x, y, v, w\in U\). By Lemma 1, it gives either \([y, w]=0\) or \(\alpha ^{-1}(g(w))=0\) for all \(w, y\in U\). By using similar arguments as we have used above to get \(g=0\).

Using similar approach we conclude that the same result holds for \(G(uv)-F(u)F(v)\in Z(R)\) for all \(u, v\in U\). This completes the proof. \(\square \)

Theorem 2

Let R be a 2-torsion free prime ring, U a nonzero square-closed Lie ideal of R. Suppose that G and F are two generalized \((\alpha , \alpha )\)-derivations associated with \((\alpha , \alpha )\)-derivations g and d respectively on R. If \(G(uv)\pm F(u)F(v)\pm \alpha (uv) \in Z(R)\) for all \(u, v \in U\), then either \(d=g=0\) or \(U\subseteq Z(R)\).

Proof

If F is a generalized \((\alpha , \alpha )\)-derivation on R associated with \((\alpha , \alpha )\)-derivation d, then \(F\pm \alpha \) is also a generalized \((\alpha , \alpha )\)-derivation on R associated with \((\alpha , \alpha )\)-derivation d on R. By putting \(G=G\pm \alpha \) in Theorem 1, we get required result. \(\square \)

The following corollary is an immediate consequence of Theorems 1 and 2.

Corollary 1

Let R be a 2-torsion free prime ring, U a nonzero square-closed Lie ideal of R. Suppose that F is a generalized \((\alpha , \alpha )\)-derivation associated with \((\alpha , \alpha )\)-derivation d on R. If one of the following holds:

  1. (i)

    \(F(uv)\pm \alpha (uv) \in Z(R)\);

  2. (ii)

    \(F(u)F(v)\pm \alpha (uv) \in Z(R)\);

  3. (iii)

    \(F(uv)\pm F(u)F(v)\in Z(R)\),

for all \(u, v \in U\), then either \(d=0\) or \(U\subseteq Z(R)\).

Theorem 3

Let R be a 2-torsion free prime ring, U a nonzero square-closed Lie ideal of R. Suppose that G and F are two generalized \((\alpha , \alpha )\)-derivations associated with \((\alpha , \alpha )\)-derivations g and d respectively on R. If \(G(uv)\pm F(u)F(v)\pm \alpha (vu)=0\) for all \(u, v \in U\), then \(U\subseteq Z(R)\).

Proof

Suppose on contrary that \(U \not \subseteq Z(R)\). We begin with the situation

$$\begin{aligned} G(uv)\pm F(u)F(v)+\alpha (vu)=0 \end{aligned}$$
(3.9)

for all \(u, v \in U\). Replacing v by 2vw in (3.9), and using 2-torsion freeness of R, we get

$$\begin{aligned} G(uv)\alpha (w)+\alpha (uv)g(w)\pm F(u)F(v)\alpha (w)\pm F(u)\alpha (v)d(w)+\alpha (vwu)=0\qquad \quad \end{aligned}$$
(3.10)

for all \(u, v, w \in U\). Using (3.9), (3.10) gives

$$\begin{aligned} \alpha (uv)g(w)\pm F(u)\alpha (v)d(w)+\alpha (v[w, u])=0 \end{aligned}$$
(3.11)

for all \(u, v, w \in U\). Putting \(v=2yv\) in (3.11), we obtain

$$\begin{aligned} \alpha (uyv)g(w)\pm F(u)\alpha (yv)d(w)+\alpha (yv[w, u])=0 \end{aligned}$$
(3.12)

for all \(u, v, w, y \in U\). Again replacing u by 2uy in (3.11), we obtain

$$\begin{aligned} \alpha (uyv)g(w)\pm F(u)\alpha (yv)d(w)\pm \alpha (u)d(y)\alpha (v)d(w)+\alpha (v[w, uy])=0 \end{aligned}$$
(3.13)

for all \(u, v, w, y \in U\). Subtracting (3.12) from (3.13), we get

$$\begin{aligned} \alpha (u)d(y)\alpha (v)d(w)\pm \alpha (v[w, uy]-yv[w, u])=0 \end{aligned}$$
(3.14)

for all \(u, v, w, y \in U\). Again replacing u by 2wu in (3.14), we obtain

$$\begin{aligned} \alpha (wu)d(y)\alpha (v)d(w)\pm \alpha (v[w, wuy]-yv[w, wu])=0 \end{aligned}$$
(3.15)

for all \(u, v, w, y \in U\). Left multiplying (3.14) by \(\alpha (w)\) and then subtracting from (3.15), we get \(\alpha \Big (vw[w, uy]-yv[w, wu]-wv[w, uy]+wyv[w, u]\Big )=0\), which implies that

$$\begin{aligned}{}[v, w][w, uy]-[yv, w][w, u]=0 \end{aligned}$$
(3.16)

for all \(u, v, w, y \in U\). In particular for \(u=w\), it implies that \([v, w]w[w, y]=0\) for all \(v, w, y\in U\). Putting \(v=2uv\) in last expression, we get \([u, w]vw[w, y]=0\) for all \(v, w, y\in U\). In particular \(y=u\), we obtain \([u, w]vw[w, u]=0\) for all \(u, v, w\in U\). By primeness of R, \(w[w, u]=0\) for all \(u, w\in U\). It implies that \([w, u]=0\) for all \(u, w\in U\). By Lemma 2, a contradiction. \(\square \)

Theorem 4

Let R be a 2-torsion free prime ring, U a nonzero square-closed Lie ideal of R. Suppose that G and F are two generalized \((\alpha , \alpha )\)-derivations associated with \((\alpha , \alpha )\)-derivations g and d respectively on R. If any one of the following holds:

  1. (i)

    \(G(uv)\pm F(u)F(v)\pm \alpha ([u, v])=0\);

  2. (ii)

    \(G(uv)\pm F(u)F(v)\pm \alpha (u\circ v)=0\),

for all \(u, v \in U\), then \(U\subseteq Z(R)\).

The following corollary is a particular case of Theorems 3 and 4.

Corollary 2

Let R be a 2-torsion free prime ring, U a nonzero square-closed Lie ideal of R. Suppose that F is a generalized \((\alpha , \alpha )\)-derivation associated with \((\alpha , \alpha )\)-derivation d on R. If any one of the following holds:

  1. (i)

    \(F(uv)\pm \alpha ([u, v])=0\);

  2. (ii)

    \(F(u)F(v)\pm \alpha (u\circ v)=0\);

  3. (iii)

    \(F(uv)\pm \alpha (vu)=0\);

  4. (iv)

    \(F(u)F(v)\pm \alpha (vu)=0\),

for all \(u, v \in U\), then \(U\subseteq Z(R)\).

We conclude this paper by giving an example which shows that the primeness of the ring in our results can not be dropped.

Example 1

Consider S be a set of integers. Suppose that

$$\begin{aligned}R = \left\{ \left( \begin{array}{c@{\quad }c@{\quad }c} 0 &{} x &{} y \\ 0 &{} 0 &{} z \\ 0 &{} 0 &{} 0\\ \end{array} \right) \mid x, y, z \in S \right\} \end{aligned}$$

and \(\alpha \) is mapping on R such that \(\alpha \left( \begin{array}{c@{\quad }c@{\quad }c} 0 &{} x &{} y \\ 0 &{} 0 &{} z \\ 0 &{} 0 &{} 0\\ \end{array} \right) = \left( \begin{array}{c@{\quad }c@{\quad }c} 0 &{} -x &{} y \\ 0 &{} 0 &{} -z \\ 0 &{} 0 &{} 0\\ \end{array} \right) \) for all \(x, y, z\in S\). Define \(G, g:R\longrightarrow R\) as

$$\begin{aligned} G\left( \begin{array}{c@{\quad }c@{\quad }c} 0 &{} x &{} y \\ 0 &{} 0 &{} z \\ 0 &{} 0 &{} 0\\ \end{array} \right) = \left( \begin{array}{c@{\quad }c@{\quad }c} 0 &{} 0 &{} x \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0\\ \end{array} \right) , g\left( \begin{array}{cccc} 0 &{} x &{} y \\ 0 &{} 0 &{} z \\ 0 &{} 0 &{} 0\\ \end{array} \right) = \left( \begin{array}{cccc} 0 &{} 0 &{} z \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0\\ \end{array} \right) . \end{aligned}$$

It is easy to verify that G is a generalized \((\alpha , \alpha )\)-derivation on R associated with \((\alpha , \alpha )\)-derivation g on R. Again define mappings F and d on R such that

$$\begin{aligned}F\left( \begin{array}{c@{\quad }c@{\quad }c} 0 &{} x &{} y \\ 0 &{} 0 &{} z \\ 0 &{} 0 &{} 0\\ \end{array} \right) = \left( \begin{array}{cccc} 0 &{}-x &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0\\ \end{array} \right) \textit{ and } d\left( \begin{array}{cccc} 0 &{} x &{} y \\ 0 &{} 0 &{} z \\ 0 &{} 0 &{} 0\\ \end{array} \right) = \left( \begin{array}{cccc} 0 &{} 0 &{} -y \\ 0 &{} 0 &{} z \\ 0 &{} 0 &{} 0\\ \end{array} \right) \end{aligned}$$

respectively. We notice that F is a generalized \((\alpha , \alpha )\)-derivation associated with a \((\alpha , \alpha )\)-derivation d on R. Let \(U= \left\{ \left( \begin{array}{c@{\quad }c@{\quad }c} 0 &{} x &{} y \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0\\ \end{array} \right) \mid x, y \in S \right\} \). Here we see that U is a square closed Lie ideal of R and satisfying the following conditions;

(i) \(G(XY)\pm F(X)F(Y)\in Z(R)\), (ii) \(G(XY)\pm F(X)F(Y)+\alpha (XY)\in Z(R)\),

(iii) \(G(XY)\pm \alpha (XY)\in Z(R)\), (iv) \(F(X)F(Y)\pm \alpha (XY) \in Z(R)\), (v) \(F(XY)\pm F(X)F(Y)\in Z(R)\) for all \(X, Y \in U\) but \(U\not \subseteq Z(R)\). Let \(X_1 = \left( \begin{array}{c@{\quad }c@{\quad }c} 0 &{} a &{} a \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0\\ \end{array} \right) \) with \(a \ne 0 \in S\), we note that \(X_1RX_1=0\) but \(X_1 \ne 0\) implies that R is not a prime ring. In this example we see that \(U\not \subseteq Z(R)\). Hence the primeness of hypothesis is essential in Theorems 1, 2 and Corollary 1.