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Action of higher derivations on semiprime rings.
- Source :
-
Georgian Mathematical Journal . Jun2024, p1. 13p. - Publication Year :
- 2024
-
Abstract
- 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 <italic>d</italic> of <italic>R</italic> 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. [ABSTRACT FROM AUTHOR]
- Subjects :
- *QUOTIENT rings
*RING theory
*PRIME ideals
*INTEGERS
Subjects
Details
- Language :
- English
- ISSN :
- 1072947X
- Database :
- Academic Search Index
- Journal :
- Georgian Mathematical Journal
- Publication Type :
- Academic Journal
- Accession number :
- 178041875
- Full Text :
- https://doi.org/10.1515/gmj-2024-2026