Find a Injection from the set of all Ideals of $L/I$ to the set of ideals of $I$ where $L$ is a Lie-algebra and $I\subset L$ an ideal

Question

Since I am new to Lie-algebras and my algebra knowledges are a bit rusty. I am not sure how to approach this task. I got the definition, that $I$ is a Ideal, if $[x,y]$ for $x\in I$,$y\in L$. What do we know in general about ideals of the quotient $L/I$ and of ideals of ideals,rsp. $I$? How does a element of $L/I$ even look like (is it just like $x+i$ where $x\in L,i\in I$)? Is there even a way to count ideals given a finite field? I hope my questions are not that incomprehensible.

Answer 1

This is not true, take $L$ be any non zero nilpotent Lie algebra (or any Lie algebra which has more than one ideal) and $I=\{0\}$ the zero ideal.

I believe you want to find an injection between the set of ideals of $L/I$ and the set of ideals of $L$ which contain $I$ in this case if $p:L\rightarrow L/I$ is the quotient map the map which associates to an ideal of $L/I$ its inverse image by $p$ realizes this injection.