  In this paper, we compare the abelian subalgebras and ideals of maximal
dimension for finite-dimensional Zinbiel algebras. We study Zinbiel algebras
containing maximal abelian subalgebras of codimension $1$ and supersolvable
Zinbiel algebras in which such subalgebras have codimension $2$, and we also
analyze the case of filiform Zinbiel algebras. We give examples to clarify some
results, including listing the values for $\alpha$ and $\beta$ for the low
dimensional Zinbiel algebras over the complex field that have been classified.

Rings and Algebras