The commutativity of prime rings with homoderivations

Let 𝑅 be a ring with center 𝑍(𝑅) , and 𝐼 be a nonzero left ideal. An additive mapping ℎ:𝑅 → 𝑅 is called a homoderivation on 𝑅 if ℎ(𝑥𝑦) = ℎ(𝑥)ℎ(𝑦) + ℎ(𝑥)𝑦 + 𝑥ℎ(𝑦) for all 𝑥. 𝑦 ∈ 𝑅 . In this paper, we prove the commutativity of R if any of the following conditions is satisfied for all 𝑥.𝑦 ∈ 𝑅 : (i) 𝑥ℎ(𝑦) ± 𝑥𝑦 ∈ 𝑍(𝑅). (ii) 𝑥ℎ(𝑦) ± 𝑦𝑥 ∈ 𝑍(𝑅). (iii) 𝑥ℎ(𝑦) ± [𝑥. 𝑦] ∈ 𝑍(𝑅) (iv)[𝑥.𝑦] ∈ 𝑍(𝑅)(v)[ℎ(𝑥)𝑦] ± 𝑥𝑦 𝑍(𝑅)and (vi) [ℎ(𝑥). 𝑦] ± 𝑦𝑥 ∈ 𝑍(𝑅) . This result is in the sprite of the well-known theorem of the commutativity of prime and semiprime rings with derivations satisfying certain polynomial constraints. Also, we prove that the commutativity of prime ring on R , if R admits a nonzero homoderivation ℎ such that ℎ([𝑥. 𝑦]) = ±[𝑥. 𝑦] for all 𝑥. 𝑦 in a nonzero left ideal.


Introduction
*Throughout, denotes a ring with a center ( ). We write [ . ] − and is called the commutator. A ring is called prime if = 0 implies = 0 = 0 and is called semiprime if = 0 then = 0. A derivation on is an additive mapping : → satisfying ( ) = ( ) + ( ) for all . ∈ . El Sofy (2000) defined a homoderivation on to be an additive mapping ℎ from into itself such that ℎ( ) = ℎ( )ℎ( ) + ℎ( ) + ℎ( ) for all . ∈ . The only additive map which is both derivation and homoderivation on prime ring is the zero map. If ⊆ , then a mapping : → preserves ( ) ⊆S. A mapping : → is said to be zero-power valued on if preserves and if for each ∈ , there exists a positive integer ( ) > 1 such that ( ) = 0 (El Sofy, 2000). Ashraf and Rehman (2001) had shown that if is a prime ring, an ideal of and : → is a derivation of , then is a commutative ring if and only if satisfies any one of the properties Motivated by these results, we prove a similar result regarding homoderivations. To achieve our aim, we will use the following lemma.

Lemma 1.1 (Lemma 4):
Let b and ab be in the center of a prime ring R. If b≠0, then a is in Z(R) (Mayne, 1984).

Remark 1.2 (Remark 3):
Let R be a prime ring. If R contains a nonzero commutative left ideal, then R is a commutative ring (Bresar, 1993). Lemma 1.3 (Lemma 1.1): Let R be a ring and 0≠I a right ideal of R. Suppose that a∈I such that a n = 0 for a fixed integer n. Then R has a nonzero nilpotent ideal (Herstein, 1969).

Lemma 1.4 (Corollary 2.5):
Let R be a prime ring of characteristic not 2 and I a nonzero left ideal. If R admits a nonzero homoderivation h which is centralizing on I, then R is commutative.

On the commutative conditions
Theorem 2.1: Let R be a prime ring of characteristic not 2, and I be a nonzero left ideal in R. If h is a nonzero homderivation which is zero-power valued on I. Then, for all x, y∈I, the following conditions are equivalent: vii. is commutative.
To prove (ii) → (vii). By hypothesis, we have Replace x by yx in (3), we have y(xh(y) ± yx) ∈ Z(R). By Lemma 1.1, y ∈ Z(R) or xh(y) ± yx = 0, If y ∈ Z(R) for all y ∈ I. hence I ⊆ Z(R). Therefore I is commutative. By Remark 1.2, R is commutative. If Replace y by xy in (4) we have xh(x)(h(y) ± y) = 0 for all x. y ∈ I. Since h is zero-power valued on I , so xh(x)y = 0 for all x. y ∈ I. Hence, xh(x)RI = 0 for all x ∈ I ⋅ By primeness of R, we get xh(x) = 0 for all x ∈ I. So, by (3) we get x 2 = 0 for all x ∈ I. By Lemma 1.3, this is contradiction. To prove (iii) → (vii). By hypothesis, we have Replace x by yx in (5), ( ℎ( ) ± [ . ]) ∈ ( ) either ∈ ( ) or ℎ( ) ± [ . ] = 0. If ∈ ( ) and ⊆ ( ) then I is commutative ideal. By Remark 1.2, R is commutative. If Replace y by yx in (6) we get: Since h is zero-power valued on I. so we get xyh(x) = 0 for all x. y ∈ I which implies xRIh(x) = 0 for all x ∈ I. By primeness of R either x = 0 or Ih(x) = 0. But I ≠ 0. So Ih(x) = 0 for all x ∈ I. Form (6) we have [x. y] = 0 for all x. y ∈ I. Then I is commutative ideal. By Remark 1.2. We have R is commutative. To prove (iv) ⟶ (vii) by hypothesis we get: Replace x by xy in (7) (ℎ( ) ± [ . ]) ∈ ( ) Replace y by xy in (8), h(x)(h(y) ± y)x = 0. Since h is zero-power valued on I, so h(x)yx = 0 for all x. y ∈ I . Then we get h(x)RIx = 0.
By primeness of R we have h(x) = 0 since Ix ≠ 0 for all x ∈ I.
If h(x) = 0 for all x ∈ I , we have [x. y] = 0 by (8), then I is commutative. By Remark 1.2, R is commutative.
To prove (v) ⟶ (vii). By hypothesis, we have Replace y by yh(x) in (8)   By primeness of R and I ≠ 0, we have [h(x). x] = 0 for all x ∈ I. By Lemma 1.4, R is commutative.

By Lemma 1.1 either h(x) ∈ Z(R) or [h(x). y] + yx = 0.
If h(x) ∈ Z(R) for all x ∈ I. then [h(x). x] = 0. By Lemma 1.4, R is commutative. If Since I is a nonzero left ideal [h(x). x]RI = 0. By primeness of R and I ≠ 0, we have [h(x). x] = 0 for all x ∈ I. By Lemma 1.4, R is commutative. Daif and Bell (1992) Lemma 3.1 (Corollary 3.4.2): Let R be a prime ring of char(R) ≠ 2. and I ≠ (0), a two sides ideal of R. If R admits a a nonzero homoderivation h on I such that h([x. y]) = [x. y] for all x. y ∈ I. Then R is commutative (El Sofy, 2000).

On condition [ . ] = −[ . ]
Theorem 3.2: Let I be nonzero left ideal in a prime ring R that admits a homoderivation h which is zeropower valued on I satisfying xy + h(xy) = yx + h(yx) for all x. y ∈ I. Then R is commutative.
Theorem 3: Let R be a prime ring with char (R) ≠ 2 and I be a nonzero ideal of R. Suppose h is a nonzero homoderivation which is zero-power valued on I. If one of the following conditions are satisfied for all x. y ∈ I: i. ℎ( ) = .
Then R is commutative.
By Theorem (3.2), we obtain R is commutative.

Conclusion
The goal of this paper is to prove the commutativity of prime rings with homoderivation which satisfying some algebraic conditions. This article is divided into two sections; in the first section, the commutativity of prime rings R was proved of the homoderivation on R satisfies following conditions for all In the second section, we investigate the commutativity of prime ring, if R admits a nonzero homoderivation h such that h([x. y]) = ± [x. y] for all x. y in a nonzero left ideal.