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Let be a prime ring, with no non-zero nil right ideal, a non-zero drivation of , a non-zero two-sided ideal of . If, for any , , there exists such that , then is commutative. As a consequence we extend the result to Lie ideals.
Let be a prime ring of characteristic different from 2, its right Martindale quotient ring and its extended centroid. Suppose that , are generalized skew derivations of with the same associated automorphism , and is a non-central polynomial over such that
for all . Then there exists such that for all .
Let be a prime ring of characteristic different from 2 and 3, its right Martindale quotient ring, its extended centroid, a non-central Lie ideal of and a fixed positive integer. Let be an automorphism of the ring . An additive map is called an -derivation (or a skew derivation) on if for all . An additive mapping is called a generalized -derivation (or a generalized skew derivation) on if there exists a skew derivation on such that for all . We prove that, if ...
Let be a prime ring of characteristic different from 2, be its right Martindale quotient ring and be its extended centroid. Suppose that is a non-zero generalized skew derivation of and f(x₁,..., xₙ) is a non-central multilinear polynomial over with n non-commuting variables. If there exists a non-zero element a of such that a[ (f(r₁,..., rₙ)),f(r₁, ..., rₙ)] = 0 for all r₁, ..., rₙ ∈ , then one of the following holds:
(a) there exists λ ∈ such that (x) = λx for all x ∈ ;
(b) there exist and...
Let be a prime ring of characteristic different from , the Utumi quotient ring of , the extended centroid of , a non-central Lie ideal of , a non-zero generalized derivation of . Suppose that for all , then one of the following holds: (1) there exists such that for all ; (2) satisfies the standard identity and there exist and such that for all . We also extend the result to the one-sided case. Finally, as an application we obtain some range inclusion results of...
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