Generalized (σ,τ) higher derivations in prime rings

Let R be a ring and U be a Lie ideal of R. Suppose that σ, τ are endomorphisms of R. A family D = {d_n}_{n ∈ N}of additive mappings d_n:R → R is said to be a (σ,τ)- higher derivation of U into R if d₀ = I_R, the identity map on R and holds for all a, b ∈ U and for each n ∈ N. A family F = {f_n}_{n ∈ N}of additive mappings f_n:R → R is said to be a generalized (σ,τ)- higher derivation (resp. generalized Jordan (σ,τ)-higher derivation) of U into R if there exists a (σ,τ)- higher derivation D = {d_n}_{n ∈ N}of U into R such that, f₀ = I_R and (resp. holds for all a, b ∈ U and for each n ∈ N. It can be easily observed that every generalized (σ,τ)-higher derivation of U into R is a generalized Jordan (σ,τ)-higher derivation of U into R but not conversely. In the present paper we shall obtain the conditions under which every generalized Jordan (σ,τ)- higher derivation of U into R is a generalized (σ,τ)-higher derivation of U into R.

Throughout (σ,τ) the present paper R will denote an associative ring with center Z(R). For any x,y ∈ R denote the commutator xy − yx by [x,y]. Recall that a ring R is prime if aRb = {0} implies that a = 0 or b = 0. An additive subgroup U of R is said to be a Lie ideal of R if [U,R] ⊆ U. A Lie ideal U of R is said to be a square closed Lie ideal of R if u²∈U for all u∈U. An additive mapping d:R→R is said to be a derivation (resp. Jordan derivation) of R if d(xy) = d(x)y + xd(y)(resp. d(x²) = d(x)x + xd(x)) holds for all x,y ∈ R. Now let D = {d_n}_{n ∈ N} be a family of additive mappings d_n:R. Following Hasse and Schimdt (→ R1937), D is said to be a higher derivation (resp. Jordan higher derivation) on R if d₀ = I_R(the identity map on R) and (resp. holds for all a, b ∈ R and for each n ∈ N. Several interesting results on higher derivation can be seen in Haetinger (2000). Let σ,τ be endomorphisms of R. An additive mapping d:R→R is said to be a (σ,τ)-derivation (resp. Jordan (σ,τ)-derivation) of R if d(xy) = σ(x)d(y) + d(x)τ(y) (resp. d(x²) = σ(x)d(x) + d(x)τ(x)) holds for all x,y ∈ R. For a fixed a ∈ R, the map d_a:R→Rgiven by d_a(x) = aτ (x) − σ(x)a for all x ∈ R is a (σ,τ)- derivation which is said to be a (σ,τ)-inner derivation determined by a.

Inspired by the notion of (σ,τ)- derivation the authors together with Haetinger (2010) introduced the concept of a (σ,τ)- higher derivation as follows: A family D = {d_n}_{n ∈ N} of additive mappings d_n:R → R is said to be a (σ,τ)- higher derivation of R if d₀ = I_R and holds for all a, b ∈ R and for each n ∈ N (If U is a Lie ideal of R, then D is said to be a (σ,τ)- higher derivation of U into R if the corresponding conditions are satisfied for all a, b ∈ U).

Following Bres̆ar (1991), an additive mapping F:R→Ris said to be a generalized derivation if there exists a derivation d:R→R such that F(xy) = F(x)y + xd(y) holds for all x, y ∈ R. Motivated by the definition of generalized derivation the notion of generalized higher derivation was introduced by Cortes and Haetinger (2005). A family F = {f_n}_{n ∈ N}of additive maps f_n:R→Ris said to be a generalized higher derivation of R if there exists a higher derivation D = {d_n}_{n ∈ N} of R such that f₀ = I_R and for all a, b ∈ R and for each n ∈ N.

An additive mapping F:R→R is said to be a generalized (σ,τ)-inner derivation if F(x) = σ(x)b + aτ(x) holds for some fixed a, b ∈ R and for all x ∈ R. If F is a generalized (σ,τ)-inner derivation, a simple computation yields that F(xy) = σ(x)d_−b(y) + F(x)τ(y), where d_−b is a (σ,τ)-inner derivation. With this point of view an additive mapping F:R→R is said to be a generalized (σ,τ)-derivation on R if there exists a (σ,τ)-derivation d:R→R such that F(xy) = σ(x)d(y) + F(x)τ(y) holds for all x,y ∈ R. For such an example let S be any ring and . Define an additive map F:R→R such that and endomorphisms σ,τ:R→R such that and . Then it can be easily seen that F is a generalized (σ,τ)-derivation on R with associated (σ,τ)- derivation d:R→R such that .

In view of the above definition we introduce the analogue of (σ,τ)-higher derivation in a more general setting.

Let σ,τ be endomorphims of R. A family F = {f_n}_{n ∈ N} of additive maps f_n:R→R is said to be generalized (σ,τ)-higher derivation (resp. generalized Jordan (σ,τ)-higher derivation) of R if there exists a (σ,τ)-higher derivation D = {d_n}_{n ∈ N} of R such that f₀ = I_R and (resp. holds for all a, b ∈ R and for each n ∈ N.

Let U be a Lie ideal of R. Then F is said to be a generalized (σ,τ)-higher derivation (resp. generalized Jordan (σ,τ)-higher derivation) of U into R if the above corresponding conditions are satisfied for all a, b ∈ U.

Example 1.1. Let R be an algebra over the field of rationals and σ,τ be the endomorphisms of R. Define , for all n ∈ N, where f is a generalized (σ,τ)- derivation on R with associated (σ,τ)-derivation δsuch that fσ = σf and δτ = τδ(the above example of generalized (σ,τ)-derivation ensures the existence of such f). Consider the sequence {F_n}_{n ∈ N}, this defines a generalized (σ,τ)- higher derivation with associated (σ,τ)-higher derivation .

If we choose the underlying f to be a generalized Jordan (σ,τ)−derivation on R which is not a generalized (σ,τ)−derivation on R then one can easily construct an example of a generalized Jordan (σ,τ)− higher derivation on R which is not a generalized (σ,τ)− higher derivation on R.

It is easy to see that every derivation on R is a Jordan derivation but the converse need not be true in general. A well-known result due to Herstein (2002) states that every Jordan derivation on a prime ring of characteristic different from two is a derivation. This result was further generalized by many authors in various directions (see Ashraf et al. 2001; Bres̆ar and Vukman 1991where further references can be found). Motivated by these results Ferrero and Haetinger (2002) generalized Herstein’s theorem for higher derivations and proved that every Jordan higher derivation on a prime ring of characteristic different from two is a higher derivation. The authors together with Haetinger (Ashraf et al. 2010) further generalized the above result in the setting of (σ,τ)-higher derivation of R. The main objective of the present paper is to find the conditions on R under which every generalized Jordan (σ,τ)-higher derivation of R is a generalized (σ,τ)-higher derivation of R. In fact our results generalize, extend and compliment several results obtained earlier in this direction.

Recently, Haetinger (2002) proved that if R is a prime ring of characteristic different from 2 and U a square closed Lie ideal such that U⫅̸Z(R). Then every Jordan higher derivation of U into R is a higher derivation of U into R. The following theorem shows that the above result still holds for arbitrary square closed Lie ideal of R that is, U may be central.

Theorem 2.1. Let R be a prime ring such that char(R) ≠ 2 and U be a square closed Lie ideal of R. Suppose that σ,τ are endomorphisms of R such that στ = τσ and τ is one-one & onto. Then every generalized Jordan (σ,τ)-higher derivation of U into R is a generalized (σ,τ)-higher derivation of U into R.

In order to develop the proof of the theorem, we begin with the following known lemma:

Lemma 2.1. ((Ferrero and Haetinger 2002), Lemma 2.3) Assume that R is a 2-torsion free prime ring and U a square closed Lie ideal of R such that U⫅̸Z(R). Let G₁,G₂,⋯,G_nbe additive groups, S:G₁×G₂×⋯×G_n→R and T:G₁×G₂×⋯×G_n→ R be the mappings which are additive in each argument. If S(a₁,a₂,⋯,a_n)xT(a₁,a₂,⋯,a_n) = 0 for every x ∈ U, a_i∈G_i, i = 1,2,⋯,n then S(a₁,a₂,⋯,a_n) = 0 for every a_i∈G_i, i = 1,2,⋯,nor T(b₁,b₂,⋯,b_n) = 0 for every b_i∈G_i, i = 1,2,⋯,n.

Lemma 2.2. Let R be a ring and σ, τ be endomorphisms of R such that στ = τσ and F = {f_n}_{n ∈ N}be a generalized Jordan (σ,τ)-higher derivation of U into R with associated (σ,τ)-higher derivation D = {d_n}_{n ∈ N} of U into R. Then for all u, v, w ∈ U and each fixed n ∈ N we have for all u, v, w ∈ U.

(i)

If R is a 2-torsion free ring then,

(ii)

(iii)

Proof. (i) For u, v ∈ U, n ∈ N we have, .By linearizing the above relation on u we obtain

for all u, v ∈ U. Again;

for all u, v ∈ U.Comparing the above expressions and reordering the indices we obtain the required result.

(ii) Since uv + vu = (u + v)²−u²−v²∈U, using (i) and replacing v by uv + vu we find that,

On the other hand,

Comparing the equations (2.1) and (2.2) and reordering the indices and using the fact that R is 2-torsion free we get the required result.

(iii) Linearizing the above result, we have

Again,

Comparing (2.3) & (2.4) and using the fact that R is 2-torsion free we get the required result.

For every fixed n ∈ N and for each u,v∈U we denote by Φ_n(u,v) the element of R such that . It is straight forward to see that if Φ_n(u,v) = 0, then F = {f_n}_{n ∈ N} is a generalized (σ,τ)-higher derivation of U into R. Trivially, by using Lemma 2.2(i) we get Φ_n(u,v) = −Φ_n(v,u), for all u,v∈U.

It can also be seen that the function Φ_nis additive in both the arguments.

Lemma 2.3. Let R be a 2-torsion free ring and σ,τbe endomorphisms of R such that στ = τσ. Let F = {f_n}_{n ∈ N}be a generalized Jordan (σ,τ)-higher derivation of U into R with associated (σ,τ)-higher derivation D = {d_n}_{n ∈ N}of U into R. If Φ_m(u,v) = 0, for each m<n and for all u,v∈U, then

(i) Φ_n(u,v)τⁿ[u,v] = 0, for all u,v∈U

(ii) Φ_n(u,v)τⁿ(w)τⁿ[u,v] = 0, for all u,v,w∈U.

Proof. 

(i) Since for any u,v∈U, uv + vu∈U and uv−vu∈U, we find that 2uv∈U. Suppose β = 4(uv(uv) + (uv)vu)∈U. Using Lemma 2.2(iii) we have,

Using the fact that Φ_m(u,v) = 0 for all m<n we have,

On the other hand,

Comparing (2.5) with (2.6) for f_n(β)we get Φ_n(u,v) × [τⁿ(u),τⁿ(v)] = 0 for all u, v ∈ U.

(ii) Let χ = 4(uvwvu + vuwuv) for u, v, w ∈ U. Then by Lemma 2.2(ii) we obtain

Again consider,

Applying Lemma 2.2(iii), we have

Equating (2.7) & (2.8) and using the fact that R is 2 torsion free we find that

Initially calculating the first term we have

Using the hypothesis that Φ_m(u,v) = 0 for all m < n.

Similarly the second term reduces to

Now, subtracting the equation (2.11) from (2.10) and using the hypothesis that στ = τσ we obtain

Similarly, the difference of the last two terms of equation (2.9) yields

Thus, (2.9) becomes Φ_n(u,v)τⁿ(w)τⁿ[u,v] = 0 for all u, v ∈ U.

We are now well equipped to prove our main theorem.

Proof of Theorem 2.1. Suppose that U is commutative. If U is commutative then U is also central (see the proof of Lemma 1.3 of (Herstein 1969)). We’ll proceed by induction on n. For n = 1, every generalized Jordan (σ,τ)-higher derivation reduces to generalized Jordan (σ,τ)-derivation and hence using Lemma 2.2(iii) of (Ashraf et al. 2001), we have

As U is commutative, in view of Lemma 2.2(i) of (Ashraf et al. 2001) we have

Since, d(uv) = d(vu) = σ(v)d(u) + d(v)τ(u), the above equation can be rewritten as

Comparing the equations (2.12) and (2.13) we obtain

As w is central and since τ is one-one and onto hence τ(w) is central but the center of a prime ring is free from zero divisors, the above equation implies that Φ₁(u,v) = 0 for all u, v ∈ U. Let Φ_m(u,v) = 0 for all u, v ∈ U and each m < n then from Lemma 2.2(iii), we have

for all u, v, w ∈ U.

Again by using Lemma 2.2(i) and commutativity of U, we get

Since U is commutative we find that d_j(τ^n−j(uv)) = d_j(τ^n−j(vu)) and as d_j is a (σ,τ)-higher derivation the above equation can be written as

Comparing equations (2.15) and (2.14) we find that

Since Φ_m(u,v) = 0 for all u, v ∈ U, m < n, the above equation reduces to

Since w is central and as τ is one-one and onto, τⁿ(w) is also central. But the center of a prime ring is free from the zero divisors, equation (2.16) implies that Φ_n(u,v) = 0 for all u, v ∈ U and each n ∈ N.

Now consider the possibility that U is non-commutative hence U⫅̸Z(R). Using Lemma 2.3(ii) we have Φ_n(u,v)τⁿ(w)τⁿ[u,v] = 0 for all u,v,w∈U. Since τ is one-one and onto, the relation yields that τ⁻ⁿ(Φ_n(u,v))w[u,v] = 0 for all u,v,w∈U. Hence by Lemma 2.1, τ⁻ⁿ(Φ_n(u,v)) = 0 for all u,v∈U or [u,v] = 0 for all u,v∈U. But since U is non-commutative, we find that Φ_n(u,v) = 0, for all u,v∈U and each n ∈ N.This completes the proof of our theorem.

As a consequence of the above result we find the following corollaries. Corollary 2.1 settle the conjecture given in (Ashraf et al. 2004) for a square closed Lie ideal while Corollary 2.2 improves the main Theorems of (Ashraf et al. (2010), Bres̆ar and Vukman (1988); Haetinger (2002)). □

Corollary 2.1. Let R be 2 torsion free prime ring and U a square closed Lie ideal of R. Suppose that θ,ϕ are endomorphisms of R such that ϕ is one-one, onto. If F:R→R is a generalized Jordan (θ,ϕ) derivation on U then F is a generalized (θ,ϕ) derivation on U.

Corollary 2.2. Let R be a prime ring such that char(R) ≠ 2 and U a square closed Lie ideal of R. Then every Jordan higher derivation of U into R is a higher derivation of U into R.

In the above theorem if the underlying ring is arbitrary, then we can prove the following:

Theorem 2.2. Let R be a 2- torsion free ring and U be a square closed Lie ideal of R. Suppose that σ,τ are endomorphisms of R such that στ = τσ and τ is one-one & onto. If U has a commutator which is not a right zero divisor, then every generalized Jordan (σ,τ)-higher derivation of U into R is a generalized (σ,τ)-higher derivation of U into R.

Proof. Let x,y∈U be the fixed elements such that c[x,y]=0⇒c = 0 for every c∈R. We’ll prove the result by induction on n.

We know that for n = 0, Φ₀(u,v) = 0. Hence proceeding by induction we can assume that Φ_m(u,v) = 0 for all m<n. Using Lemma 2.3(i) we have

The above equation implies that τ⁻ⁿ(Φ_n(u,v))[u,v] = 0 for all u, v ∈ U. Hence in particular, τ⁻ⁿ(Φ_n(x,y))[x,y] = 0. This implies that,

Replacing u by u + x in (2.17) we get

Again replacing v by y in (2.19) and using (2.18) we get Φ_n(u,y)[τⁿ(x),τⁿ(y)] = 0, for every u ∈ U,i.e., τ⁻ⁿ(Φ_n(u,y))[x,y] = 0. This yields that

Replace v by v + y in (2.19), to get

Replacing u by x in (2.21) and using (2.18) we obtain Φ_n(x,v)[τⁿ(x),τⁿ(y)] = 0 for all v ∈U. This yields that

Combining (2.20), (2.21) and (2.22) we find that Φ_n(u,v)[τⁿ(x),τⁿ(y)] = 0 i.e., τ⁻ⁿ(Φ_n(u,v))[x,y] = 0. Hence, we conclude that Φ_n(u,v) = 0 for all u,v∈U.This completes the proof of our theorem. □

Some special cases of the above theorem are already known and are of great interest.

Corollary 2.3. ((Ashraf et al. 2004), Theorem 2.3) Let R be a 2 torsion free prime ring and U a square closed Lie ideal of R. Suppose that σ,τ are endomorphisms of R such that τ is one-one, onto. Suppose further that U has a commutator which is not a zero divisor. If F:R→R is a generalized Jordan (σ,τ) derivation of U into R then F is a generalized (σ,τ) derivation of U into R.

Corollary 2.4. ((Cortes and Haetinger 2005), Theorem 1.3) Let R be a 2 torsion free ring such that R has a commutator which is not a right divisor and U a square closed Lie ideal of R. Then every generalized Jordan higher derivation of U into R is a generalized higher derivation of U into R.

The authors declare that they have no competing interests.

Both the authors, viz. MA and AK, with the consultation of each other, carried out this work and drafted the manuscript together. Both the authors read and approved the final manuscript.

The authors wish to thank the referees for their useful suggestions.

This research is partially supported by a grant from the Department of Science & Technology, New Delhi (Grant no. SR/S4/MS:556/08)

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