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On near-ring ideals with (σ, τ)-derivation
(2007) Gölbaşi, Öznur; Aydın, Neşet
Let N be a 3-prime left near-ring with multiplicative center Z, a (?, ?)-derivation D on N is defined to be an additive endomorphism satisfying the product rule D(xy) = ?(x)D(y) + D(x)?(y) for all x, y ? N, where ? and ? are automorphisms of N. A nonempty subset U of N will be called a semigroup right ideal (resp. semigroup left ideal) if U N ? U (resp. NU ? U) and if U is both a semigroup right ideal and a semigroup left ideal, it be called a semigroup ideal. We prove the following results: Let D be a (?, ?)-derivation on N such that ?D = D?, ?D = D?. (i) If U is semigroup right ideal of N and D(U) ? Z then N is commutative ring, (ii) If U is a semigroup ideal of N and D2(U) = 0 then D = 0. (iii) If a ? N and [D(U),a]?? = 0 then D(a) = 0 or a ? Z.
Some results for generalized lie ideals in prime rings with derivation II
(2001) Kaya, Kâzim; Gölbaşi, Öznur; Aydin, Neşet
Let R be a prime ring of characteristic different from two, d : R ? R a non-zero derivation, and M a non-zero left ideal of R. We prove the following results: (1) if a ? R and [d(R), a]?, ? = 0, then ? (a) + ?(a) ? Z, the center of R, (2) if d([R, a]?, ?) = 0, then ? (a)+?(a) ? Z, (3) if ([R,M]?, ?, a)?, ? = 0, then a ? Z, (4) d(R), a) = 0 if, and only if, d((R, a)) = 0.