MATEMATIQKI VESNIK

60 (2008), 1–2

UDK 512.558

originalni nauqni rad research paper

CENTRALIZING AND COMMUTING GENERALIZED DERIVATIONS ON PRIME RINGS

Asif Ali and Tariq Shah

Abstract. Let R be a prime ring and d a derivation on R. If f is a generalized derivation on R such that f is centralizing on a left ideal U of R, then R is commutative.

A ring R is said to be prime if aRb = 0 implies that either a = 0 or b = 0. An additive mapping d: R → R is said to be a derivation if d (xy) = d (x) y + xd (y) for all x, y ∈ R. A mapping f is said to be commuting on a left ideal U of R if [f (x) , x] = 0 for all x ∈ U and f is said to be centralizing if [f (x) , x] ∈ Z (R) for all x ∈ U. There has been considerable interest in commuting, centralizing and related mappings in prime and semiprime rings (see [2] for a partial bibliography).

In this note we extend some results of Bell and Martindale [1] for generalized derivations. An additive mapping f: R → R is said to be a generalized derivation on R if f (xy) = f (x) y + xd (y) for all x, y ∈ R (where d is a derivation on R). These mappings were introduced in [3].

Throughout this note R will represent a prime ring with Z (R) being its centre. In the following we state a well known fact as remark 1. For a nonzero element a ∈ Z (R), if ab ∈ Z (R), then b ∈ Z (R). In order to prove the main result, we find it necessary to establish the following

Lemma.

Lemma 1. If f is an additive mapping from R to R such that f is centralizing on a left ideal U of R, then for all x ∈ U ∩ Z (R), f (x) ∈ Z (R).

Proof. Since f is centralizing on U, we have [f (x + y) , x + y] ∈ Z (R), for all x, y ∈ U. This implies that

[f (x) , y] + [f (y) , x] ∈ Z (R) .  (1)

AMS Subject Classification: 16N60, 16S20

Keywords and phrases: Prime rings; commuting and centralizing maps; generalized derivations.

Now if x ∈ Z (R), then from equation (1), [f (x) , y] ∈ Z (R). Replacing y by f (x) y, we get f (x) [f (x) , y] ∈ Z (R). If [f (x) , y] = 0, then f (x) ∈ C_{R} (U), the centralizer of U in R, and hence ([1, Identity IV]) belongs to Z (R). But on the other hand if [f (x) , y] ≠ 0, it again follows from Remark 1 that f (x) ∈ Z (R).

Next we prove the result which generalizes [1, Theorem 4].

Theorem 1. Let R be a prime ring. Let d: R → R be a nonzero derivation and f be a generalized derivation on a left ideal U of R. If f is commuting on U then R is commutative.

Proof. Since f is commuting on U, we have [f(x), x] = 0 for all x ∈ U. Replacing x by x + y, we get [f (x) , y] + [f (y) , x] = 0. Now by substituting y = yx and simplifying we arrive at [yd (x) , x] = 0. Replacing y by ry, we get [r, x] U d (x) = 0 for all x ∈ U and r ∈ R. Since R is a prime ring, therefore either [r, x] = 0 or d (x) = 0 for all r ∈ R. So for any element x ∈ U, either x ∈ Z(R) or d(x) = 0. Since d is nonzero on R, then by [4, Lemma 2], d is nonzero on U. Suppose d(x) ≠ 0, for some x ∈ U, then x ∈ Z(R). Suppose z ∈ U is such that z ∉ Z(R), then d (z) = 0 and x + z ∉ Z(R). This implies d (x + z) = 0 and so d (x) = 0, a contradiction. This implies z ∈ Z(R) for all z ∈ U. Thus U is commutative and hence by [4, Lemma 3], R is commutative.

Now we are ready to prove the result which involves centralizing generalized derivations on left ideals containing central elements.

Theorem 2. Let U be a left ideal of a prime ring R such that U ∩ Z(R) ≠ 0. Let d be a nonzero derivation and f be a generalized derivation on R such that f is centralizing on U. Then R is commutative.

Proof. We assume that Z(R) ≠ 0 because otherwise f is commuting on U and there is nothing left to prove. Now for a nonzero z ∈ Z (R), we replace x by zy in (1) and get [f (z) , y]y+z[d (y) , y]+z[f (y) , y] ∈ Z (R). Now by Lemma 1, f (z) ∈ Z (R) and therefore z[d (y) , y] + z[f (y) , y] ∈ Z (R). But as f is centralizing on U, we have z[f (y) , y] ∈ Z (R) and consequently z[d (y) , y] ∈ Z (R). Since z is nonzero, it follows from Remark 1 that [d (y) , y] ∈ Z (R). This implies d is centralizing on U and hence by [1, Theorem 4], we conclude that R is commutative.

REFERENCES

[1] H. E. Bell and W. S. Martindale, III Centralizing mappings of semiprime rings, Canad. Math. Bull. 30 (1987), 92–101.

[2] M. Bresar, Centralizing mappings and derivations in prime rings, J. Algebra 156 (1993), 385–394.

[3] B. Hvala, Generalized derivation in rings, Comm. Algebra, 26(4) (1998), 1147–1166.

[4] J. H. Mayne, Centralizing mappings of prime rings, Canad. Math. Bull., 27(1) (1984), 122– 126.

(received 23.11.2005)

Department of Mathematics, Quaid-i-Azam University, Islamabad-Pakistan

E-mail: dr asif ali@yahoo.com, stariqshah@gmail.com