THE COMMUTATIVITY OF SEMIPRIME RINGS WITH SYMMETRIC Bi-(α, α)-DERIVATIONS

Abstract

Let R be a semiprime ring, I a nonzero ideal of R, D : R × R → R a symmetric bi-(α, α)-derivation, d be the trace of D and α an automorphism. In the present paper, we shall prove that R contains a nonzero central ideal if any one of the following holds: i) d([x, y] α,α) ± [x,y]α,α∈ Cα,α,ii)[d(x), d(y)]α,α± [x, y]α,α∈ Cα,α, iii)d((x ◦ y)α,α) ± (x ◦ y)α,α∈ Cα,α,iv)(d(x) ◦ d(y))α,α± (xoy)α,α∈ Cα,α, v)d((x ◦ y)α,α) ± [x, y]α,α∈ Cα,α,vi)(d(x) ◦ d(y))α,α± [x, y]α,α∈ Cα,α, vii)d([x, y]α,α) ± (x ◦ y)α,α∈ Cα,α,viii)[d(x), d(y)]α,α± (x ◦ y)α,α∈ C
α,α, ix)d (x) d (y) ± [x, y]α,α∈ Cα,α,x)d (x) d (y)±(x ◦ y)α,α∈ Cα,α, xi)[d (x) , y]α,α∈ Cα,α, xii)d [x, y]α,α±[d(x), y]α,α∈ Cα,α,xiii)d(x ◦ y)α,α ± [d(x),y] ∈ Cα,α, for all x, y ∈ I.