Generalized Derivations in Rings with Involution

Dar, Nadeem Ahmad; Khan, Abdul Nadim

doi:10.1142/s1005386717000244

Cited by 8 publications

(5 citation statements)

References 17 publications

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“…In 2014, Ali et al [1] studied the SCP maps in different way on rings possessing involution. They established the commutativity of prime ring of characteristic not two possessing second kind of involution satisfying [φ(x), φ(x * )] − [x, x * ] = 0 for every x ∈ R, where φ is a nonzero derivation of R. Later, Dar and Khan [11] improved this result by studying the case of generalized derivations. Further, Khan and Ali [16] studied SCP maps as endomorphisms on rings with involution.…”

Section: Results On Prime Idealsmentioning

confidence: 99%

Actions of generalized derivations on prime ideals in $*$-rings with applications

Abbasi¹,

Khan

2023

Hacettepe Journal of Mathematics and Statistics

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In this paper, we make use of generalized derivations to scrutinize the deportment of prime ideal satisfying certain algebraic $*$-identities in rings with involution. In specific cases, the structure of the quotient ring $\mathscr{R}/\mathscr{P}$ will be resolved, where $\mathscr{R}$ is an arbitrary ring and $\mathscr{P}$ is a prime ideal of $\mathscr{R}$ and we also find the behaviour of derivations associated with generalized derivations satisfying algebraic $*$-identities involving prime ideals. Finally, we conclude our paper with applications of the previous section's results.

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Section: Results On Prime Idealsmentioning

confidence: 99%

Actions of generalized derivations on prime ideals in $*$-rings with applications

Abbasi¹,

Khan

2023

Hacettepe Journal of Mathematics and Statistics

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“…Corollary 4.11. [13,Theorem 2.3] Let R be a 2−torsion free noncommutative prime ring with involution of the second kind. If R admits a nonzero generalized derivation F associated with a derivation d such that [F(x), F(x * )] = [x, x * ] for all x ∈ R, then F(x) = x for all x ∈ R or F(x) = −x for all x ∈ R. Corollary 4.12.…”

Section: Examplesmentioning

confidence: 99%

“…Further, Ali et al [4] studied strong commutativity preserving type derivations in rings with involution. Recently, Dar and Khan [13] proved the following result in this domain: let R be a noncommutative 2−torsion free prime ring with involution of the second kind. If R admits a generalized derivation F : R → R associated with a derivation d :…”

Section: Introductionmentioning

confidence: 99%

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A classification of generalized derivations in rings with involution

Bhushan

Sandhu²,

Ali

et al. 2021

Filomat

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Let R be a ring. An additive mapping F : R ? R is called a generalized derivation if there exists a derivation d of R such that F(xy) = F(x)y + xd(y) for all x,y ? R. The main purpose of this paper is to characterize some specific classes of generalized derivations of rings. Precisely, we describe the structure of generalized derivations of noncommutative prime rings with involution that belong to a particular class of generalized derivations. Consequently, some recent results in this line of investigation have been extended. Moreover, some suitable examples showing that the assumed hypotheses are crucial, are also given.

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“…Ali et al [4] studied strong commutativity preserving like derivations in rings with involution. Recently, Dar and Khan [12] proved the following: Let R be a non-commutative 2-torsion free prime ring with involution of the second kind. If R admits a generalized derivation F : R → R associated with a derivation…”

Section: Introductionmentioning

confidence: 99%