1. Annihilator of Power Values of Generalized Derivations in Prime Rings
- Author
-
Giovanni Scudo and Shuliang Huang
- Subjects
Physics ,General Mathematics ,010102 general mathematics ,Center (category theory) ,General Physics and Astronomy ,General Chemistry ,01 natural sciences ,Prime (order theory) ,010101 applied mathematics ,Combinatorics ,Annihilator ,Identity (mathematics) ,Prime ring ,General Earth and Planetary Sciences ,Ideal (ring theory) ,0101 mathematics ,General Agricultural and Biological Sciences ,Quotient ring - Abstract
Let R be a prime ring of characteristic different from 2, U its Utumi quotient ring, C the center of U, F a non-zero generalized derivation of R and L a non-commutative Lie ideal of R. Suppose that there exists $$0\ne a\in R$$ such that $$a(u^s[F(u),u]u^t)^n=0$$ for all $$u \in L$$ , where $$s\ge 0, t\ge 0, n\ge 1$$ are fixed integers. Then either $$F(x)=\alpha x$$ for all $$x\in R$$ with $$\alpha \in C$$ or R satisfies $$s_4(x_1,\ldots ,x_4)$$ , the standard identity in four variables, and $$F(x)=bx+xb+\alpha x$$ for all $$x\in R$$ , for some $$b\in U$$ and $$\alpha \in C$$ .
- Published
- 2017