On the Variety of Zero Divisors in Algebras

Fabrizio Colombo, Irene Sabadini and Daniele C. Struppa
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Fabrizio Colombo: Politecnico di Milano, Dipartimento di Matematica
Irene Sabadini: Politecnico di Milano, Dipartimento di Matematica
Daniele C. Struppa: Chapman University, Donald Bren Presidential Chair in Mathematics

Chapter Chapter 4 in Michele Sce's Works in Hypercomplex Analysis, 2020, pp 57-68 from Springer

Abstract: Abstract This chapter contains the translation of the paper: M. Sce, Sulla varietà dei divisori dello zero nelle algebre, (Italian) Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 23 (1957), 39–44 as well as some comments and historical remarks. In this work, after some brief consideration on matrices on (skew) fields, we show how the study of the variety of zero divisors in alternative algebras on fields of characteristic different from 2 and in Jordan algebras on characteristic zero fields, can be led to the study of these varieties in simple, central algebras. We then show that the dimension of the variety of zero divisors in alternative algebras on fields of characteristic different from 2 is given by the order of the algebra minus the order of the smallest primitive algebra factor of the simple components of the algebra.

Date: 2020
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DOI: 10.1007/978-3-030-50216-4_4

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