Higher derivations and Posner’s second theorem for semiprime rings

Prajapati, B.

doi:10.1007/s11565-020-00352-4

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Herstein's Results For Higher Derivations1

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2024

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Cited by 2 publications

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“…Tus, d n (x) ∈ C for all x ∈ Q. Since R forms a subring of Q, we conclude d n (x) ∈ Z(R) for all x ∈ R. Hence, [d n (x), x] � 0 for all x ∈ R. Te application of ( [12], Teorem 3.1) implies either R is commutative or some linear combination of (d i )'s maps center to zero. Hence, this is the required result.…”

Section: Herstein's Results For Higher Derivationsmentioning

confidence: 80%

Higher Derivations Satisfying Certain Identities in Rings

Alali,

Ali,

Rafiquee

et al. 2024

Journal of Mathematics

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Section: Herstein's Results For Higher Derivationsmentioning

confidence: 80%

Higher Derivations Satisfying Certain Identities in Rings

Alali,

Ali,

Rafiquee

et al. 2024

Journal of Mathematics

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Action of higher derivations on semiprime rings

Ali,

Varshney

2024

Georgian Mathematical Journal

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Let m , n {m,n} be the fixed positive integers and let ℛ {\mathcal{R}} be a ring. In 1978, Herstein proved that a 2-torsion free prime ring ℛ {\mathcal{R}} is commutative if there is a nonzero derivation d of R such that [ d ⁢ ( ϱ ) , d ⁢ ( ξ ) ] = 0 {[d(\varrho),d(\xi)]=0} for all ϱ , ξ ∈ R {\varrho,\xi\in R} . In this article, we study the above mentioned classical result for higher derivations and describe the structure of semiprime rings by using the invariance property of prime ideals under higher derivations. Precisely, apart from proving some other important results, we prove the following. Let ( d i ) i ∈ ℕ {(d_{i})_{i\in\mathbb{N}}} and ( g j ) j ∈ ℕ {(g_{j})_{j\in\mathbb{N}}} be two higher derivations of semiprime ring ℛ {\mathcal{R}} such that [ d n ⁢ ( ϱ ) , g m ⁢ ( ξ ) ] ∈ Z ⁢ ( ℛ ) {[d_{n}(\varrho),g_{m}(\xi)]\in Z(\mathcal{R})} for all ϱ , ξ ∈ ℐ {\varrho,\xi\in\mathcal{I}} , where ℐ {\mathcal{I}} is an ideal of ℛ {\mathcal{R}} . Then either ℛ {\mathcal{R}} is commutative or some linear combination of ( d i ) i ∈ ℕ {(d_{i})_{i\in\mathbb{N}}} sends Z ⁢ ( ℛ ) {Z(\mathcal{R})} to zero or some linear combination of ( g j ) j ∈ ℕ {(g_{j})_{j\in\mathbb{N}}} sends Z ⁢ ( ℛ ) {Z(\mathcal{R})} to zero. We enrich our results with examples that show the necessity of their assumptions. Finally, we conclude our paper with a direction for further research.

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Higher derivations and Posner’s second theorem for semiprime rings

Cited by 2 publications

References 14 publications

Higher Derivations Satisfying Certain Identities in Rings

Higher Derivations Satisfying Certain Identities in Rings

Action of higher derivations on semiprime rings

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