  The goal of this work is to study the ideals of the Goldman Lie algebra $S$.
To do so, we construct an algebra homomorphism from $S$ to a simpler algebraic
structure, and focus on finding ideals of this new structure instead. The
structure $S$ can be regarded as either a $\mathbb{Q}$-module or a
$\mathbb{Q}$-module generated by free homotopy classes. For $\mathbb{Z}$-module
case, we proved that there is an infinite class of ideals of $S$ that contain a
certain finite set of free homotopy classes. For $\mathbb{Q}$-module case, we
can classify all the ideals of the new structure and consequently obtain a new
class of ideals of the original structure. Finally, we show an interesting
infinite chain of ideals that are not those ideals obtained by considering the
new structure.
