I'm stuck on the next problem from Humphreys: Let $L$ be nilpotent. Prove that L has an ideal of codimension 1.
Here is my attempt: I have that $L\neq[L,L]$ (so $[L,L]\subsetneq L$), so $\dim(L/[L,L])\geq 1$. If $\dim(L/[L,L])=1$ then the result follows by taking $I=[L,L]$. Now suppose $\dim(L/[L,L])=n>1$.
Let $\beta=\text{Span}\{e_{1},\ldots,e_{n}\}$ be a basis for $L/[L,L]$. Since $L/[L,L]$ is abelian, then $I=\text{Span}\{e_{1},\ldots,e_{n-1}\}$ is an ideal of $L/[L,L]$ with codimension 1. By the correspondence theorem, I corresponds to an ideal of $L$, say $K$, such that $[L,L]\subset K$ (furhtermore: $[L,L]\subset K\subset L$). So, I'm stuck here, how can I show that K has codimension 1?
Also, if we consider $\pi:L\rightarrow L/[L,L]$ the canonical projection, I think that $\pi^{-1}(I)$ has codimension 1 (I know that is an ideal of $L$), but a can't see.
I will apreciate any hint, thanks.
HINT:
Consider a subspace $K$ such that $[L,L]\subset K$ and $K$ of codimension $1$. It turns out that $K$ is an ideal! Indeed: $[L,K] \subset [L,L]\subset K$.
About your question on codimensions: if $p \colon V \to W$ is a surjective linear map and $W'\subset W$ is a subspace then the codimension of $p^{-1}(W')$ in $V$ is the codimension of $W'$ in $W$. That is some linear algebra.