Rendiconti del Circolo Matematico di Palermo Series 2, 2019
Let R be a non commutative prime ring of characteristic different from 2 with Utumi quotient ring... more Let R be a non commutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C . Let d and $$\delta $$ δ be two derivations of R and S be the set of evaluations of a multilinear polynomial $$f(x_1,\ldots ,x_n)$$ f ( x 1 , … , x n ) over C which is not central valued. Let $$p,q\in R$$ p , q ∈ R . We prove the followings. (1) If $$pud\delta (u)+\delta d(u)uq=0$$ p u d δ ( u ) + δ d ( u ) u q = 0 for all $$u\in S$$ u ∈ S and $$p+q\notin C$$ p + q ∉ C . Then either $$d=0$$ d = 0 or $$\delta =0$$ δ = 0 . (2) If $$pud(u)+d(u)uq=0$$ p u d ( u ) + d ( u ) u q = 0 for all $$u\in S$$ u ∈ S . Then either $$d=0$$ d = 0 or $$p=q\in C$$ p = q ∈ C , $$d(x)=[a,x]$$ d ( x ) = [ a , x ] for some $$a\in U$$ a ∈ U and $$f(x_1,\ldots ,x_n)^2$$ f ( x 1 , … , x n ) 2 is central valued.
Rendiconti del Circolo Matematico di Palermo Series 2, 2019
Let R be a non commutative prime ring of characteristic different from 2 with Utumi quotient ring... more Let R be a non commutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C . Let d and $$\delta $$ δ be two derivations of R and S be the set of evaluations of a multilinear polynomial $$f(x_1,\ldots ,x_n)$$ f ( x 1 , … , x n ) over C which is not central valued. Let $$p,q\in R$$ p , q ∈ R . We prove the followings. (1) If $$pud\delta (u)+\delta d(u)uq=0$$ p u d δ ( u ) + δ d ( u ) u q = 0 for all $$u\in S$$ u ∈ S and $$p+q\notin C$$ p + q ∉ C . Then either $$d=0$$ d = 0 or $$\delta =0$$ δ = 0 . (2) If $$pud(u)+d(u)uq=0$$ p u d ( u ) + d ( u ) u q = 0 for all $$u\in S$$ u ∈ S . Then either $$d=0$$ d = 0 or $$p=q\in C$$ p = q ∈ C , $$d(x)=[a,x]$$ d ( x ) = [ a , x ] for some $$a\in U$$ a ∈ U and $$f(x_1,\ldots ,x_n)^2$$ f ( x 1 , … , x n ) 2 is central valued.
Uploads
Papers by Balchand Prajapati