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Journal of Lie Theory 31 (2021), No. 4, 991--1002
Copyright Heldermann Verlag 2021

On Certain Classes of Algebras in which Centralizers are Ideals

Ripan Saha
Department of Mathematics, Raiganj University, Raiganj 733134, India

David A. Towers
Department of Mathematics and Statistics, Lancaster University, Lancaster, England

This paper is primarily concerned with studying finite-dimensional anti-commutative nonassociative algebras in which every centralizer is an ideal. These are shown to be anti-associative and are classified over a field F of characteristic different from 2; in particular, they are nilpotent of class at most 3 and metabelian. These results are then applied to show that a Leibniz algebra over a field of charactersitic zero in which all centralizers are ideals is solvable.

Keywords: Anti-commutative algebra, anti-associative algebra, Lie algebra, Leibniz algebra, mock-Lie algebra, centralizer, nilpotent algebra.

MSC: 17A30, 17A32, 17B30.

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