Journals / Turkish Journal of Mathematics / 2013 / Cilt: 37 - Sayı: 2
Generalized derivations of prime rings on multilinear polynomials with annihilator conditions
- Pages
- 231–243
- DOI
- —
Abstract
Let K be a commutative ring with unity, R be a prime K-algebra with characteristic not 2, U be the right Utumi quotient ring of R, C the extended centroid of R, I a nonzero right ideal of R and a a fixed element of R. Let g be a generalized derivation of R and f(X1,..., Xn) a multilinear polynomial over K. If ag(f(x1,...,xn))f(x1,...,xn)=0 for all x1,...,xn \in I, then one of the following holds: (1) aI=ag(I)=0; (2) g(x)=bx+[c,x] for all x\in R, where b,c\in U. In this case either [c,I]I=0=abI or aI=0=a(b+c)I; (3) [f(X1,...,Xn),Xn+1]Xn+2 is an identity for I.
Özet
Let K be a commutative ring with unity, R be a prime K-algebra with characteristic not 2, U be the right Utumi quotient ring of R, C the extended centroid of R, I a nonzero right ideal of R and a a fixed element of R. Let g be a generalized derivation of R and f(X1,..., Xn) a multilinear polynomial over K. If ag(f(x1,...,xn))f(x1,...,xn)=0 for all x1,...,xn \in I, then one of the following holds: (1) aI=ag(I)=0; (2) g(x)=bx+[c,x] for all x\in R, where b,c\in U. In this case either [c,I]I=0=abI or aI=0=a(b+c)I; (3) [f(X1,...,Xn),Xn+1]Xn+2 is an identity for I.