A note on a pair of derivations of semiprime rings

Abstract

We study certain properties of derivations on semiprime rings. The main purpose is to prove the following result: let R be a semiprime ring with center Z(R), and let f , g be derivations of R such that f (x)x + xg(x) ∈ Z(R) for all x ∈ R, then f and g are central.As an application, we show that noncommutative semisimple Banach algebras do not admit nonzero linear derivations satisfying the above central property. We also show that every skew-centralizing derivation f of a semiprime ring R is skew-commuting.2000 Mathematics Subject Classification: 47A50, 47B50. Introduction and preliminaries. Throughout, R denotes a ring with center Z(R).We write [x, y] for xy − yx. We will frequently use the identities Every central mapping is obviously commuting but not conversely, in general. A lot of work has been done on centralizing mappings (see, e.g., [3,4,5] and the references therein). A mapping f : R → R is called skew-centralizing if f (x)x + xf (x) ∈ Z(R)for all x ∈ R; in particular, if f (x)x + xf (x) = 0 for all x ∈ R, then it is called skew-commuting. We denote the radical of a Banach algebra A by rad(A).We now recall some facts concerning semiprime rings and their extended centroids. For any semiprime ring R, one can construct the ring of quotients Q of R [1]. As R can be embedded isomorphically in Q, we consider R as a subring of Q. If the element q ∈ Q commutes with every element in R, then q belongs to C, the center of Q. C contains the centroid of R and is called the extended centroid of R. In general, C is a von Neumann regular ring, and it is a field if and only if R is a prime [1, Theorem 5]. For more information on extended centroid of R, we refer to [2].Brešar [6, Theorem 2] has proved that if R is a prime ring of characteristic not 2 and f : R → R is an additive skew-commuting mapping (i.e., f satisfies f (x)x + xf (x) = 0 for all x ∈ R), then f = 0.Moreover, Brešar [5, Theorem 4.1] has considered a pair of derivations on a prime ring and has proved the following. Let R be a prime ring and U a nonzero left ideal of R. Suppose that the derivations d and g of R are such that d(u)u − ug(u) ∈ Z(R) for all u ∈ U . If d ≠ 0, then R is commutative.

On ideals of prime rings involving $n$-skew commuting additive mappings with applications

“…Further, Chaudhary and Thaheem [18] extended the above mentioned results for semiprime rings and showed that if R is a semiprime ring and f, g a pair of derivations of R such that f (x)x + xg(x) ∈ Z(R), ∀ x ∈ R, then f and g are central. Inspired by these work's, Ali et al [1] established the following result.…”

Approximate Derivations with the Radical Ranges of Noncommutative Banach Algebras

“…Employing the same argument in the proof of the previous corollaries, we see that 1 and 2 are linear derivations. Chaudhry and Thaheem's result [20] guarantees that 1 (A) ⊆ (A) and 2 (A) ⊆ (A). This implies that 1 and 2 are centralizing mappings.…”

“…Then, we have by (22) From now on, we suppose that U := { ∈ C : | | = 1}. (2) and (20), then maps A into the intersection of its center (A) and its radical rad (A).…”

“…In this paper, we will establish the stability of functional inequalities with ring derivations and will deal with the problem for the radical range of these functional inequalities on Banach algebra by considering the base of noncommutative versions for the result of Singer and Wermer: Mathieu and Murphy [18] verify that every continuous centralizing linear derivation on a Banach algebra maps into the radical and Brešar [19] proved that every centralizing linear derivation on a semiprime Banach algebra maps into the intersection of the center and the radical. Moreover, Chaudhry and Thaheem [20] showed that two ring derivations on semiprime ring with suitable property map into the center.…”

Pairs of Derivations on Rings and Banach Algebras

“…This theorem has been generalized by many people in different ways. Chaudhry and Thaheem [7] prove that if d, g are a pair of derivations of a semiprime ring TZ such that d(x)x+xg(x) E Cu for all χ ETZ, then d and g are both central. In particular, if TZ is a prime ring, then / = 0 and 5 = 0.…”