Abstract
Let R be an associative ring. A mapping f : R −→ R is said to be additive if f (x +y) = f (x) +f (y) holds for all x;y ∈ R: An additive mapping d : R −→ R is called a derivation if d(xy) = d(x)y + xd(y) holds for all x;y ∈ R: In this paper, we investigate commutativity of prime and semiprime rings satisfying certain identities involving additive mappings and derivations. Moreover, some results have also been discussed.
Highlights
Let R be an associative ring with center Z(R) and extended centroid C
We know that if R is a semiprime ring and I is an ideal of R, rR(I) = lR(I)
An additive mapping d : R −→ R is called a derivation if d(xy) = d(x)y + xd(y) holds for all x, y ∈ R
Summary
Let R be an associative ring with center Z(R) and extended centroid C. In [30, Theorem 2], proved that if a prime ring R admits a nonzero derivation d A number of authors have extended these theorems of Posner and Mayne; they have showed that derivations, automorphisms, and some related maps cannot be centralizing on certain subsets of noncommutative prime (and some other) rings.
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