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Some ring theory from Jenő Szigeti. (English) Zbl 1340.16019

Introduction: The present overview concentrates on three areas of Szigeti’s work. “Eulerian polynomial identities” [J. Szigeti et al., J. Algebra 161, No. 1, 90-101 (1993; Zbl 0802.16015)] deals essentially with polynomials in several non-commuting indeterminates corresponding to directed Eulerian graphs. “Lie nilpotent determinant theory” adapts to the non-commutative case the classical concepts of determinant, adjoint and characteristic polynomial to yield analogues of well known linear algebra results, especially over Lie nilpotent rings (developed in the series of papers, e.g. [J. Szigeti and Z. Tuza, Linear Multilinear Algebra 42, No. 1, 43-51 (1997; Zbl 0869.15002); J. Szigeti and L. Van Wyk, Commun. Algebra 43, No. 11, 4783-4796 (2015; Zbl 1333.16003)]). “Centralizers and zero-level centralizers” [V. Drensky et al., J. Algebra 324, No. 12, 3378-3387 (2010; Zbl 1214.16022); J. Szigeti, Isr. J. Math. 107, 229-235 (1998; Zbl 0918.16020)] is about some non-commutative extensions of theorems on centralizers and double centralizers in matrix algebras, with additional considerations of two-sided annihilators.
We aim at a condensed but self contained presentation of a selection of results.

MSC:

16R10 \(T\)-ideals, identities, varieties of associative rings and algebras
16S50 Endomorphism rings; matrix rings
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
15A15 Determinants, permanents, traces, other special matrix functions
15A30 Algebraic systems of matrices

Biographic References:

Szigeti, Jenő
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