ǂA ǂnote on derivations in semiprime rings

Joso Vukman (Author), Irena Kosi-Ulbl (Author)

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].

Keywords

matematika;asociativni kolobarji in algebre;odvajanja;polprakolobarji;ne zaključna dela;mathematics;associative rings an algebras;derivations;semiprime rings;

Data

Language:	English
Year of publishing:	2005
Typology:	1.01 - Original Scientific Article
Organization:	UM PEF - Faculty of Education
UDC:	512.552
COBISS:	14369032
ISSN:	0161-1712
Views:	1239
Downloads:	354
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	Slovenian
Secondary title:	Opazka o odvajanjih na polprakolobarjih
Secondary keywords:	Matematika;Algebra;Kolobarji (algebra);Polkolobarji;
URN:	URN:SI:UM:
Type (COBISS):	Article
Pages:	str. 3347-3350
Issue:	20
Chronology:	2005
DOI:	10.1155/IJMMS.2005.3347
ID:	10842668