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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 Lie ideals and Jordan left derivations of prime rings

Mohammad Ashraf, Nadeem-ur-Rehman (2000)

Archivum Mathematicum

Let R be a 2-torsion free prime ring and let U be a Lie ideal of R such that u 2 ∈ U for all u ∈ U . In the present paper it is shown that if d is an additive mappings of R into itself satisfying d ( u 2 ) = 2 u d ( u ) for all u ∈ U , then d ( u v ) = u d ( v ) + v d ( u ) for all u , v ∈ U .

On ordered division rings

Ismail M. Idris (2001)

Colloquium Mathematicae

Prestel introduced a generalization of the notion of an ordering of a field, which is called a semiordering. Prestel's axioms for a semiordered field differ from the usual (Artin-Schreier) postulates in requiring only the closedness of the domain of positivity under x ↦ xa² for non-zero a, in place of requiring that positive elements have a positive product. Our aim in this work is to study this type of ordering in the case of a division ring. We show that it actually behaves just as in the commutative...

On ordered division rings

Ismail M. Idris (2003)

Czechoslovak Mathematical Journal

Prestel introduced a generalization of the notion of an ordering of a field, which is called a semiordering. Prestel’s axioms for a semiordered field differ from the usual (Artin-Schreier) postulates in requiring only the closedness of the domain of positivity under x → x a 2 for nonzero a , instead of requiring that positive elements have a positive product. In this work, this type of ordering is studied in the case of a division ring. It is shown that it actually behaves the same as in the commutative...

On skew derivations as homomorphisms or anti-homomorphisms

Mohd Arif Raza, Nadeem ur Rehman, Shuliang Huang (2016)

Commentationes Mathematicae Universitatis Carolinae

Let R be a prime ring with center Z and I be a nonzero ideal of R . In this manuscript, we investigate the action of skew derivation ( δ , ϕ ) of R which acts as a homomorphism or an anti-homomorphism on I . Moreover, we provide an example for semiprime case.

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