On the structure of Leibniz algebras whose subalgebras are ideals or core-free

Abstract

An algebra \(L\) over a field \(F\) is said to be a Leibniz algebra (more precisely, a left Leibniz algebra) if it satisfies the Leibniz identity: \([[a, b], c] = [a, [b, c]] - [b, [a, c]]\) for all \(a, b, c \in L\). Leibniz algebras are generalizations of Lie algebras. A subalgebra \(S\) of a Leibniz algebra \(L\) is called a core-free, if \(S\) does not include a non-zero ideal. We study the Leibniz algebras, whose subalgebras are either ideals or core-free.