Cohomology of general coinduced representions

ABSTRACT

Let R be a prime ring with a nonzero derivation d and let f(X ₁,…,X _t ) be a multilinear polynomial over C, the extended centroid of R. Suppose that b[d(f(x ₁,…,x _t )), f(x ₁,…,x _t )] _n  = 0 for all x _i  ∈ R, where 0 ≠ b ∈ R and n is a fixed positive integer. Then f(X ₁,…,X _t ) is centrally valued on R unless char R = 2 and dim _C RC = 4. We prove a more generalized version by replacing R with a left ideal.