Structure of a quotient ring $$\pmb {R/P}$$ with generalized derivations acting on the prime ideal P and some applications

Idrissi, Moulay Abdallah; Oukhtite, L.

doi:10.1007/s13226-021-00173-x

Cited by 9 publications

(3 citation statements)

References 16 publications

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“…Numerous studies have revealed that the global structure of A is typically related to the behavior of additive mappings formed on A. Recent work has investigated the commutativity of the factor ring A/P, where P is the prime ideal of any arbitrary ring A, using algebraic identities in P, including derivations and generalized derivations (see [6][7][8][9][10][11]).…”

Section: Example 1 Let D Be Any Ring and Letmentioning

confidence: 99%

On Multiplicative (Generalized)-Derivation Involving Semiprime Ideals

Alnoghashi

Naji

Rehman

2023

Journal of Mathematics

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Let A be any arbitrary associative ring, P a semiprime ideal, and J a nonzero ideal of A . In this study, using multiplicative (generalized)-derivations, we explore the behavior of semiprime ideals that satisfy certain algebraic identities. Moreover, examples are provided to demonstrate that the restrictions imposed on the hypotheses of the various theorems are necessary.

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Section: Example 1 Let D Be Any Ring and Letmentioning

confidence: 99%

On Multiplicative (Generalized)-Derivation Involving Semiprime Ideals

Alnoghashi

Naji

Rehman

2023

Journal of Mathematics

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“…The algebra of derivations and generalized derivations play a crucial role in the study of * -functional identities and their applications. In, 2022, some work have been done by researcher on the structure of a quotient ring R/P with the help of different additive mappings (See [7,8,15]). In this paper, we are interested in the study of rings with involution given as a quotient ring R/P, where R is an arbitrary ring and P is a prime ideal of R involving certain * -differential/functional identities on prime ideals.…”

Section: Introductionmentioning

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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“…Over the last few years, the interest on this topic has been increased and numerous papers concerning these mappings on prime rings have been published (see [4,9,[30][31][32][33][34][35][36][37] and references therein). In [38], Creedon studied the action of derivations of prime ideals and proved that if e is a derivation of a ring A and L is a semiprime ideal of A such that A/L is characteristic-free and e k (L) ⊆ L, then e(L) ⊆ L for some positive integer k. Very recently, Idrissi and Oukhtite [39] investigated the structure of a quotient ring A/L via the action of generalized derivations on the prime ideal of L. For more recent works, see [40][41][42] and references therein. In view of the above observations and motivation, the aim of the present paper was to prove the following main theorems.…”

Section: Introductionmentioning

confidence: 99%

Posner’s Theorem and ∗-Centralizing Derivations on Prime Ideals with Applications

Ali,

Alsuraiheed,

Khan

et al. 2023

Mathematics

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A well-known result of Posner’s second theorem states that if the commutator of each element in a prime ring and its image under a nonzero derivation are central, then the ring is commutative. In the present paper, we extended this bluestocking theorem to an arbitrary ring with involution involving prime ideals. Further, apart from proving several other interesting and exciting results, we established the ∗-version of Vukman’s theorem. Precisely, we describe the structure of quotient ring A/L, where A is an arbitrary ring and L is a prime ideal of A. Further, by taking advantage of the ∗-version of Vukman’s theorem, we show that if a 2-torsion free semiprime A with involution admits a nonzero ∗-centralizing derivation, then A contains a nonzero central ideal. This result is in the spirit of the classical result due to Bell and Martindale (Theorem 3). As the applications, we extended and unified several classical theorems. Finally, we conclude our paper with a direction for further research.

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Structure of a quotient ring $$\pmb {R/P}$$ with generalized derivations acting on the prime ideal P and some applications

Cited by 9 publications

References 16 publications

On Multiplicative (Generalized)-Derivation Involving Semiprime Ideals

On Multiplicative (Generalized)-Derivation Involving Semiprime Ideals

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

Posner’s Theorem and ∗-Centralizing Derivations on Prime Ideals with Applications

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