A Lie ring is a triple $(G,+, [\ ,\ ]),$ where $(G,+)$ is an abelian group and $ [\ ,\ ]$ is a bilinear map satisfying
- $[x,x]=0$
- $[\ ,\ ]$ is bilinear
- $[[x,y],z]+[[y,z],x]+[[z.x],y]=0,\ \forall\ x,y,z\in G.$ (Jacobi identity).
The center of the Lie ring is defined as $$Z(G)=\{x\ |\ [x,y]=0,\ \forall\ y\in G\}.$$
Let $(G,+, [\ ,\ ])$ be a Lie ring with free presentation $$0\rightarrow R\rightarrow F\rightarrow G\rightarrow 1.$$ Now consider the quotient Lie ring $\tilde{F}=\frac{F}{R\cap [F,F]}$ and $\tilde{R}=\frac{R}{R\cap [F,F]}$. Then the subring $Z(\tilde{F})\cap [\tilde{F},\tilde{F}]$ of $Z(\tilde{F})$ has a complement subring in $Z(\tilde{F})$, i.e. there exists an ideal $\tilde{T}$ of $Z(\tilde{F})$ such that $$Z(\tilde{F})=Z(\tilde{F})\cap [\tilde{F},\tilde{F}]\times \tilde{T}.$$
My question is: $\textbf{How such an ideal $\tilde{T}$ exists?}$
I have seen in the article Isoclinism in pair of Lie rings that authors have taken as assumption that if $\tilde{x}=x+[R,F]\in\frac{F}{[R,F]} $ such that $\tilde{x}^n\in \frac{[F,F]}{[R,F]}$ for some integer $n\in \mathbb{N}$ then $\tilde{x}\in \frac{[F,F]}{[R,F]}$ and assumed that this holds. Using this assumption they claimed following two points:
$(1)$ The factor Lie ring $\frac{Z(\tilde{F})}{Z(\tilde{F})\cap [\tilde{F},\tilde{F}]}$ is isomorphic to some ideal of the free abelian Lie ring $\frac{\tilde{F}}{[\tilde{F},\tilde{F}]}$
$(2)$ There exists an ideal $\tilde{T}$ of $\tilde{F}$ such that $$Z(\tilde{F})=Z(\tilde{F})\cap [\tilde{F},\tilde{F}]\times \tilde{T}.$$
It can be seen that the second claim can be obtained from the first one, but I am not able to see the first one. Please guide me in this regard.
