Algebre, določene z ničelnim produktom : doktorska disertacija

slovenian

doktorska dela

2012

Univerza v Mariboru, Fakulteta za naravoslovje in matematiko

The central object in this thesis is a zero product determined algebra. This concept is new. The bulk of the thesis is therefore devoted to the question whether standard examples of associative, Lie, and Jordan algebras are zero product determined. In the first part we treat associative algebras. We show that the matrix algebra over any unital algebra is zero product determined. Next we deal with multiadditive maps satisfying a certain zero product preserver condition. Their form is described on the subring generated with all idempotents of the ring in question. This result implies that every unital algebra generated with idempotents is zero product determined. We also show that every simple finite dimensional algebra, which is not a division algebra, is zero product determined. The second chapter is devoted to Lie algebras. We show that if an associative algebra B is zero Lie product determined, then so is the matrix algebra over B. By an example we justify the assumption that B must be zero Lie product determined. We conclude this part by showing that Lie algebras of skew symmetric matrices with respect to either transpose or symplectic involution are zero product determined. Next we turn our attention to Jordan algebras. We prove that the algebra of matrices over a unital algebra endowed with the Jordan product, the Jordan algebra of symmetric matrices with respect to either transpose or symplectic involution, the Albert algebra, and the Jordan algebra determined with a nondegenerate symmetric bilinear form are all zero product determined. At the end we deal with additive maps between prime rings.

08.08.2023