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Free actions on semiprime rings

Muhammad Anwar Chaudhry, Mohammad S. Samman (2008)

Mathematica Bohemica

We identify some situations where mappings related to left centralizers, derivations and generalized ( α , β ) -derivations are free actions on semiprime rings. We show that for a left centralizer, or a derivation T , of a semiprime ring R the mapping ψ R → R defined by ψ ( x ) = T ( x ) x - x T ( x ) for all x ∈ R is a free action. We also show that for a generalized ( α , β ) -derivation F of a semiprime ring R , with associated ( α , β ) -derivation d , a dependent element a of F is also a dependent element of α + d . Furthermore, we prove that for a centralizer f and...

Free associative algebras, noncommutative Gröbner bases, and universal associative envelopes for nonassociative structures

Murray R. Bremner (2014)

Commentationes Mathematicae Universitatis Carolinae

First, we provide an introduction to the theory and algorithms for noncommutative Gröbner bases for ideals in free associative algebras. Second, we explain how to construct universal associative envelopes for nonassociative structures defined by multilinear operations. Third, we extend the work of Elgendy (2012) for nonassociative structures on the 2-dimensional simple associative triple system to the 4- and 6-dimensional systems.

G r - ( 2 , n ) -ideals in graded commutative rings

Khaldoun Al-Zoubi, Shatha Alghueiri, Ece Y. Celikel (2020)

Commentationes Mathematicae Universitatis Carolinae

Let G be a group with identity e and let R be a G -graded ring. In this paper, we introduce and study the concept of graded ( 2 , n ) -ideals of R . A proper graded ideal I of R is called a graded ( 2 , n ) -ideal of R if whenever r s t ∈ I where r , s , t ∈ h ( R ) , then either r t ∈ I or r s ∈ G r ( 0 ) or s t ∈ G r ( 0 ) . We introduce several results concerning g r - ( 2 , n ) -ideals. For example, we give a characterization of graded ( 2 , n ) -ideals and their homogeneous components. Also, the relations between graded ( 2 , n ) -ideals and others that already exist, namely, the graded prime ideals,...

Generalized derivations associated with Hochschild 2-cocycles on some algebras

Jiankui Li, Jiren Zhou (2010)

Czechoslovak Mathematical Journal

We investigate a new type of generalized derivations associated with Hochschild 2-cocycles which was introduced by A. Nakajima. We show that every generalized Jordan derivation of this type from CSL algebras or von Neumann algebras into themselves is a generalized derivation under some reasonable conditions. We also study generalized derivable mappings at zero point associated with Hochschild 2-cocycles on CSL algebras.

Generalized derivations on Lie ideals in prime rings

Basudeb Dhara, Sukhendu Kar, Sachhidananda Mondal (2015)

Czechoslovak Mathematical Journal

Let R be a prime ring with its Utumi ring of quotients U and extended centroid C . Suppose that F is a generalized derivation of R and L is a noncentral Lie ideal of R such that F ( u ) [ F ( u ) , u ] n = 0 for all u ∈ L , where n ≥ 1 is a fixed integer. Then one of the following holds: ...

Generalized derivations with power values on rings and Banach algebras

Abderrahman Hermas, Abdellah Mamouni, Lahcen Oukhtite (2024)

Mathematica Bohemica

Let R be a prime ring and I a nonzero ideal of R . The purpose of this paper is to classify generalized derivations of R satisfying some algebraic identities with power values on I . More precisely, we consider two generalized derivations F and H of R satisfying one of the following identities: ...

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