| Title: | A note on derivations in semiprime rings |
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| Authors: | ID Vukman, Joso (Author)
ID Kosi-Ulbl, Irena (Author) |
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| Files: |  International_Journal_of_Mathematics_and_Mathematical_Sciences_2005_Vukman,_Kosi-Ulbl_A_note_on_derivations_in_semiprime_rings.pdf (1,78 MB)
MD5: A7218E0528FACDF8BE41908E0B705986
 http://www.hindawi.com/journals/ijmms/2005/637505/abs/
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| Language: | English |
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| Work type: | Article |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: | PEF - Faculty of Education
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| Abstract: | We prove in this note the following result. Let ▫$n>1$▫ be an integer and let ▫$R$▫ be an ▫$n!$▫-torsion-free semiprime ring with identity element. Suppose that there exists an additive mapping ▫$D : R \to R$▫ such that ▫$D(x^n)=\Sigma_{j^n}=1^{x^{n-j}}D(x)x^{j-1}$▫ is fulfilled for all ▫$ x \in R$▫. In this case, ▫$D$▫ is a derivation. This research is motivated by the work of Bridges and Bergen (1984). Throughout, ▫$R$▫ will represent an associative ring with center ▫$Z(R)$▫. Given an integer ▫$n > 1$▫, a ring ▫$R$▫ is said to be ▫$n$▫-torsion-free if for ▫$x \in R$▫, ▫$nx=0$▫ implies that ▫$x=0$▫. Recall that a ring ▫$R$▫ is prime if for ▫$ a,b \in R$▫, ▫$aRb=(0)$▫ implies that either ▫$a=0$▫ or ▫$b=0$▫, and is semiprime in case ▫$aRa=(0)$▫ implies that ▫$a=0$▫. An additive mapping ▫$D:R \to R$▫ is called a derivation if ▫$D(xy)=D(x)y+xD(y)$▫ holds for all pairs ▫$x,y \in R$▫ and is called a Jordan derivation in case ▫$D(x^2)=D(x)x+xD(x)$▫ is fulfilled for all ▫$x \in R$▫. Every derivation is a Jordan derivation. The converse is in general not true. A classical result of Herstein (1957) asserts that any Jordan derivation on a prime ring with characteristic different from two is a derivation. A brief proof of Herstein's result can be found in 1988 by Brešar and Vukman. Cusack (1975) generalized Herstein's result to ▫$2$▫-torsion-free semiprime rings (see also Brešar (1988) for an alternative proof). For some other results concerning derivations on prime and semiprime rings, we refer to [2, 7, 8, 9, 10]. |
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| Keywords: | mathematics, associative rings an algebras, derivations, semiprime rings |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Year of publishing: | 2005 |
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| Number of pages: | str. 3347-3350 |
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| Numbering: | Letn. 2005, št. 20 |
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| PID: | 20.500.12556/DKUM-66189  |
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| ISSN: | 0161-1712 |
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| UDC: | 512.552 |
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| ISSN on article: | 0161-1712 |
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| COBISS.SI-ID: | 14369032 |
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| DOI: | 10.1155/IJMMS.2005.3347 |
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| NUK URN: | URN:SI:UM:DK:ZBCR75HC |
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| Publication date in DKUM: | 14.06.2017 |
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| Views: | 1445 |
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| Downloads: | 393 |
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| Metadata: |    |
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| Categories: | Misc.
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