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Abstract: Let ▫$A$▫ be an algebra over a field ▫$F$▫ with ▫$\mathrm{char} (F) \ne 2$▫. If ▫$A$▫ is generated as an algebra by ▫$[[A,A],[A,A]]$▫, then for every skew-symmetric bilinear map ▫$\Phi:A \times A \to X$▫, where ▫$X$▫ is an arbitrary vector space over ▫$F$▫, the condition that ▫$\Phi(x^2,x)=0$▫ for all ▫$x \in A$▫ implies that ▫$\Phi(xy,z) +\Phi(zx,y) + \Phi(yz,x)=0$▫ for all ▫$x,y,z \in A$▫. This is applicable to the question of whether ▫$A$▫ is zero Lie product determined, and is also used in proving that a Jordan homomorphism from ▫$A$▫ onto a semiprime algebra ▫$B$▫ is the sum of a homomorphism and an antihomomorphism.
Keywords: bilinear map, zero Lie product determined algebra, derivation, Jordan derivation, Jordan homomorphism, functional identity
Published in DKUM: 18.08.2023; Views: 421; Downloads: 48
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Abstract: Let ▫${mathcal{T}}_n(mathcal{C})$▫ be the algebra of all ▫$n times n$▫ upper triangular matrices over a commutative unital ring ▫$mathcal{C}$▫. We describe the structure of Jordan homomorphisms from ▫${mathcal{T}}_n(mathcal{C})$▫ into an arbitrary algebra over ▫$mathcal{C}$▫. As an application a new proof of our recent result on Jordan derivations on ▫${mathcal{T}}_n(mathcal{C})$▫ is obtained.
Keywords: matematika, algebra zgornje trikotnih matrik, jordanski homomorfizem, jordansko odvajanje, mathematics, triangular matrix algebra, Jordan homomorphism, Jordan derivation
Published in DKUM: 10.07.2015; Views: 1285; Downloads: 109
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