Mohammad Ashraf, Nadeem-ur-Rehman
Abstract
Let R be a prime ring, I = (0) an ideal of R and d : R −→ R be a derivation of R. In the present paper it has been shown that R is commutative if and only if it satisfies any one of the properties d(xy)−xy ∈ Z(R),d(xy)+xy ∈ Z(R),d(xy)−yx∈ Z(R), d(xy)+yx∈ Z(R), d(x)d(y)−xy ∈ Z(R), and d(x)d(y)+xy ∈ Z(R), for all x,y ∈ I.
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Let R be a prime ring, I = (0) an ideal of R and d : R −→ R be a derivation of R. In the present paper it has been shown that R is commutative if and only if it satisfies any one of the properties d(xy)−xy ∈ Z(R),d(xy)+xy ∈ Z(R),d(xy)−yx∈ Z(R), d(xy)+yx∈ Z(R), d(x)d(y)−xy ∈ Z(R), and d(x)d(y)+xy ∈ Z(R), for all x,y ∈ I.
Key concepts: Prime (order theory), Classical XY model, Commutative property, Commutative ring, Mathematics, Ring (chemistry), Ideal (ethics), Combinatorics