Derivations of Prime and Semiprime Rings

Abstract

Abstract. Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and n a fixed positive integer.then R is commutative. We also examine the case where R is a semiprime ring.

On derivations in prime and semiprime rings with Banach algebras

“…In 2002 Ashraf and Rehman [3] proved that if R is a 2-torsion free prime ring, I is a nonzero ideal of R and d is a nonzero derivation of R such that d(x) • d(y) = x • y, for all x, y ∈ I, then R is commutative. Inspired by above mention results, we here generalized the result obtained in [1] and [3]. Moreover, we continue this line of investigation by examining what happens when a ring R satisfies the identity…”

“…In [3], Ashraf and Rehman proved that if R is a prime ring, I is a nonzero ideal of R and d is a nonzero derivation of R such that d(x • y) = x • y for all x, y ∈ I, then R is commutative. In [1], Argaç and Inceboz generalized the above result and they proved that: If a prime ring R admits a nonzero derivation d with the property (d(x • y)) n = x • y, for all x, y ∈ I, a nonzero ideal of R, where n is a fixed positive integer, then R is commutative.…”

A note on derivations in rings and Banach algebras

“…In 2002 Ashraf and Rehman [2,Theorem 4.4] also proved that if R is a 2-torsion free prime ring, I a nonzero ideal of R and d is a nonzero derivation of R such that d(x) • d(y) = x • y for all x, y ∈ I, then R is commutative. The present paper is motivated by the previous results and we here shall generalize the results obtained in [1] and [2]. Moreover, we continue this line of investigation by examining what happens for a ring R (or an algebra A) satisfying one of the following identities:…”

A pair of generalized derivations in prime, semiprime rings and in Banach algebras

“…We denote by [x, y] = xy − yx the simple commutator of the elements x, y ∈ R and by x • y = xy + yx the simple anti-commutator of x, y. A linear mapping d : R → R is called a derivation, if it satisfies the Leibnitz rule d(xy) = d(x)y+xd(y) for all x, y ∈ R. In particular, d is said to be an inner derivation induced by an element a ∈ R, if d(x) = [a, x] for all x ∈ R. More results about derivation can be found in [1,2,9,18,24,25].…”