Journals / Turkish Journal of Mathematics / 2008 / Cilt: 32 - Sayı: 2
Posner´s Second Theorem and an Annihilator Condition with Generalized Derivations
- Pages
- 197–211
- DOI
- —
Abstract
Let R be a prime ring of characteristic different from 2, with extended centroid C, U its two-sided Utumi quotient ring, d\neq 0 a non-zero generalized derivation of R, f(x1,..,xn) a non-central multilinear polynomial over C in n non-commuting variables, a \in R such that a[d(f(r1,..,rn)),f(r1,..,rn)]=0, for any r1,..,rn \in R. Then one of the following holds: 1. a=0; 2. there exists l \in C such that d(x)=l x, for all x \in R; 3. there exist q \in U and l \in C such that d(x)=(q+l)x+xq, for all x\in R, and f(x1,..,xn)2 is central valued on R.
Özet
Let R be a prime ring of characteristic different from 2, with extended centroid C, U its two-sided Utumi quotient ring, d\neq 0 a non-zero generalized derivation of R, f(x1,..,xn) a non-central multilinear polynomial over C in n non-commuting variables, a \in R such that a[d(f(r1,..,rn)),f(r1,..,rn)]=0, for any r1,..,rn \in R. Then one of the following holds: 1. a=0; 2. there exists l \in C such that d(x)=l x, for all x \in R; 3. there exist q \in U and l \in C such that d(x)=(q+l)x+xq, for all x\in R, and f(x1,..,xn)2 is central valued on R.