Two results concerning symmetric be-derivations on prime rings

Abstract

Naj bo ▫$R$▫ kolobar. Biaditivna simetrična preslikava ▫$D(.,.): R \times R \to R$▫ je simetrična biderivacija, če je za vsak ▫$y \in R$▫ preslikave ▫$x \mapsto D(x,y)$▫ derivacija. V članku sta dokazana dva rezultata o simetričnih biderivacijah na prakolobarjih. Prvi rezultat pravi naslednje: če sta ▫$D_1$▫ in ▫$D_2$▫ simetrični biderivaciji na prakolobarju s karakteristiko različno od dva in tri tako, da je ▫$D_1(x,x)D_2(x,x) = 0, \quad x \in R$▫, potem je ▫$D_1 = 0$▫ ali ▫$D_2 = 0$▫.

Keywords

matematika;asociativni kolobarji in algebre;kolobar;prakolobar;odvajanje;simetrično bi-odvajanje;ne zaključna dela;mathematics;associative rings and algebras;ring;prime ring;derivation;symmetric bi-derivation;

Data

Language:	English
Year of publishing:	1990
Typology:	1.01 - Original Scientific Article
Organization:	UM PEF - Faculty of Education
UDC:	512.552
COBISS:	3081220
ISSN:	0001-9054
Views:	906
Downloads:	29
Average score:	0 (0 votes)
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Other data

Secondary language:	Slovenian
Secondary title:	Dva rezultata o simetričnem odvajanju na prakolobarjih
Secondary abstract:	Let ▫$R$▫ be a ring. A bi-additive symmetric mapping ▫$D(.,.): R \times R \to R$▫ is called a symmetric bi-derivation if, for any fixed ▫$y \in R$▫, the mapping ▫$x \mapsto D(x,y)$▫ is a derivation. The purpose of this paper is to prove two results concerning symmetric bi-derivations on prime rings. The first result states that, if ▫$D_1$▫ and ▫$D_2$▫ are symmetric bi-derivations on a prime ring of characteristic different from two and three such that ▫$D_1(x,x)D_2(x,x) = 0$▫ holds for all ▫$x \in R$▫, then either ▫$D_1 = 0$▫ or ▫$D_2 = 0$▫. The second result proves that the existence of a nonzero symmetric bi-derivation on a prime ring of characteristic different from two and three, such that ▫$[[D(x,x),x],x] \in Z(R)$▫ holds for all ▫$x \in R$▫, where ▫$Z(R)$▫ denotes the center of ▫$R$▫, forces ▫$R$▫ to be commutative.
URN:	URN:SI:UM:
Type (COBISS):	Article
Pages:	str. 181-189
Volume:	ǂVol. ǂ40
Issue:	ǂiss. ǂ1
Chronology:	1990
DOI:	10.1007/BF02112294
ID:	1471798