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On a subset with nilpotent values in a prime ring with derivation

Vincenzo De Filippis (2002)

Bollettino dell'Unione Matematica Italiana

Let R be a prime ring, with no non-zero nil right ideal, d a non-zero drivation of R , I a non-zero two-sided ideal of R . If, for any x , y ∈ I , there exists n = n ⁢ x , y ≥ 1 such that d ⁢ x , y - x , y n = 0 , then R is commutative. As a consequence we extend the result to Lie ideals.

On centralizers of semiprime rings

Borut Zalar (1991)

Commentationes Mathematicae Universitatis Carolinae

Let 𝒦 be a semiprime ring and T : 𝒦 → 𝒦 an additive mapping such that T ( x 2 ) = T ( x ) x holds for all x ∈ 𝒦 . Then T is a left centralizer of 𝒦 . It is also proved that Jordan centralizers and centralizers of 𝒦 coincide.

On Jordan ideals and derivations in rings with involution

Lahcen Oukhtite (2010)

Commentationes Mathematicae Universitatis Carolinae

Let R be a 2 -torsion free * -prime ring, d a derivation which commutes with * and J a * -Jordan ideal and a subring of R . In this paper, it is shown that if either d acts as a homomorphism or as an anti-homomorphism on J , then d = 0 or J ⊆ Z ( R ) . Furthermore, an example is given to demonstrate that the * -primeness hypothesis is not superfluous.

On near-ring ideals with ( σ , τ ) -derivation

Öznur Golbaşi, Neşet Aydin (2007)

Archivum Mathematicum

Let N be a 3 -prime left near-ring with multiplicative center Z , a ( σ , τ ) -derivation D on N is defined to be an additive endomorphism satisfying the product rule D ( x y ) = τ ( x ) D ( y ) + D ( x ) σ ( y ) for all x , y ∈ N , where σ and τ 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. N U ⊂ 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 ( σ ,

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