Let R be a noncommutatiye prime ring of characteristic different from 2 with right Utumi quotient ring U and extended centroid C, I a nonzero right ideal of R. Let f(x(1), . . ., x(n)) be a non-central multilinear polynomial over C, m >= 1 a fixed integer, a a fixed element of R, G a non-zero generalized derivation of R. If aG(f(r(1), . . ., r(n)))(m) is an element of Z(R) for all r(1), . . ., r(n) is an element of I, then one of the following holds: (1) aI = aG(I) = (0); (2) G(x) = qx, for some q is an element of U and aqI = 0; (3) [f (x(1), . . ., x(n)), x(n+1)]x(n+2) is an identity for I; (4) G(x) = cx + [q, x] for all x is an element of R, where c, q is an element of U such that cI = 0 and [q, I]I = 0; (5) dim(C)(RC) <= 4; (6) G(x) = alpha x, for some alpha is an element of C; moreover alpha is an element of C and f (x(1), . . ., x(n))(m) l is central valued on R.
A Note On Certain Central Differential Identities With Generalized Derivations
DE FILIPPIS, Vincenzo
2012-01-01
Abstract
Let R be a noncommutatiye prime ring of characteristic different from 2 with right Utumi quotient ring U and extended centroid C, I a nonzero right ideal of R. Let f(x(1), . . ., x(n)) be a non-central multilinear polynomial over C, m >= 1 a fixed integer, a a fixed element of R, G a non-zero generalized derivation of R. If aG(f(r(1), . . ., r(n)))(m) is an element of Z(R) for all r(1), . . ., r(n) is an element of I, then one of the following holds: (1) aI = aG(I) = (0); (2) G(x) = qx, for some q is an element of U and aqI = 0; (3) [f (x(1), . . ., x(n)), x(n+1)]x(n+2) is an identity for I; (4) G(x) = cx + [q, x] for all x is an element of R, where c, q is an element of U such that cI = 0 and [q, I]I = 0; (5) dim(C)(RC) <= 4; (6) G(x) = alpha x, for some alpha is an element of C; moreover alpha is an element of C and f (x(1), . . ., x(n))(m) l is central valued on R.Pubblicazioni consigliate
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