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

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.