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An identity related to centralizers in semiprime rings

Joso Vukman (1999)

Commentationes Mathematicae Universitatis Carolinae

The purpose of this paper is to prove the following result: Let R be a 2 -torsion free semiprime ring and let T : R R be an additive mapping, such that 2 T ( x 2 ) = T ( x ) x + x T ( x ) holds for all x R . In this case T is left and right centralizer.

An identity with generalized derivations on Lie ideals, right ideals and Banach algebras

Vincenzo de Filippis, Giovanni Scudo, Mohammad S. Tammam El-Sayiad (2012)

Czechoslovak Mathematical Journal

Let R be a prime ring of characteristic different from 2 , U the Utumi quotient ring of R , C = Z ( U ) the extended centroid of R , L a non-central Lie ideal of R , F a non-zero generalized derivation of R . Suppose that [ F ( u ) , u ] F ( u ) = 0 for all u L , then one of the following holds: (1) there exists α C such that F ( x ) = α x for all x R ; (2) R satisfies the standard identity s 4 and there exist a U and α C such that F ( x ) = a x + x a + α x for all x R . We also extend the result to the one-sided case. Finally, as an application we obtain some range inclusion results of...

Centralizers on prime and semiprime rings

Joso Vukman (1997)

Commentationes Mathematicae Universitatis Carolinae

The purpose of this paper is to investigate identities satisfied by centralizers on prime and semiprime rings. We prove the following result: Let R be a noncommutative prime ring of characteristic different from two and let S and T be left centralizers on R . Suppose that [ S ( x ) , T ( x ) ] S ( x ) + S ( x ) [ S ( x ) , T ( x ) ] = 0 is fulfilled for all x R . If S 0 ( T ...

Centralizers on semiprime rings

Joso Vukman (2001)

Commentationes Mathematicae Universitatis Carolinae

The main result: Let R be a 2 -torsion free semiprime ring and let T : R R be an additive mapping. Suppose that T ( x y x ) = x T ( y ) x holds for all x , y R . In this case T is a centralizer.

Centralizing traces and Lie-type isomorphisms on generalized matrix algebras: a new perspective

Xinfeng Liang, Feng Wei, Ajda Fošner (2019)

Czechoslovak Mathematical Journal

Let be a commutative ring, 𝒢 be a generalized matrix algebra over with weakly loyal bimodule and 𝒵 ( 𝒢 ) be the center of 𝒢 . Suppose that 𝔮 : 𝒢 × 𝒢 𝒢 is an -bilinear mapping and that 𝔗 𝔮 : 𝒢 𝒢 is a trace of 𝔮 . The aim of this article is to describe the form of 𝔗 𝔮 satisfying the centralizing condition [ 𝔗 𝔮 ( x ) , x ] 𝒵 ( 𝒢 ) (and commuting condition [ 𝔗 𝔮 ( x ) , x ] = 0 ) for all x 𝒢 . More precisely, we will revisit the question of when the centralizing trace (and commuting trace) 𝔗 𝔮 has the so-called proper form from a new perspective. Using the aforementioned...

Commutativity theorems for rings with differential identities on Jordan ideals

L. Oukhtite, A. Mamouni, Mohammad Ashraf (2013)

Commentationes Mathematicae Universitatis Carolinae

In this paper we investigate commutativity of ring R with involution ' * ' which admits a derivation satisfying certain algebraic identities on Jordan ideals of R . Some related results for prime rings are also discussed. Finally, we provide examples to show that various restrictions imposed in the hypotheses of our theorems are not superfluous.

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