Posner´s Second Theorem and an Annihilator Condition with Generalized Derivations

Authors : Vincenzo De FILIPPIS

Pages : 197-211

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Publication Date : 0000-00-00

Article Type : Research Paper

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, finsert ignore into journalissuearticles values(x1,..,xn); a non-central multilinear polynomial over C in n non-commuting variables, a \in R such that a[dinsert ignore into journalissuearticles values(finsert ignore into journalissuearticles values(r1,..,rn););,finsert ignore into journalissuearticles values(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 dinsert ignore into journalissuearticles values(x);=l x, for all x \in R; 3. there exist q \in U and l \in C such that dinsert ignore into journalissuearticles values(x);=insert ignore into journalissuearticles values(q+l);x+xq, for all x\in R, and finsert ignore into journalissuearticles values(x1,..,xn);2 is central valued on R.
Keywords : Prime rings, derivations, left Utumi quotient rings, two sided Martindale quotient ring, differential identities