everyone. I'm studying somethings about Lie algebras and I have a simple problem (apparently), but I don't get to prove.
Question: How to prove that intersection of essential ideals of a semiprime Lie algebra is again an essential ideal?
I have proved this result only for finite intersection. I accept any suggestion. Thanks a lot.
A Lie algebra $L$ is said to be semiprime if for every non-zero ideal $I$ of $L$, $[I,I]\neq 0$. We usually denote $[I,I]$ by $I^2$.
An ideal $I$ of $L$ is said to be essential if its intersection with any non-zero ideal is again a non-zero ideal.
Let $I,J$ be essential ideals, $K$ ideal. Suppose that $I\cap J\cap K=0$. Since $I$ is essential, we have $J\cap K=0$, and in turn, since $J$ is essential, we have $K=0$. This proves that $I\cap J$ is essential (no assumption on $\mathfrak{g}$ was made).
Second, an arbitrary intersection of essential ideals in a semiprime Lie algebra can fail to be essential. Indeed, let $\mathfrak{g}$ have the basis $x$, $(y_n)_{n\ge 0}$ with $[x,y_n]=y_{n+1}$ and $[y_n,y_m]=0$. It is easy to see that the ideals are: 0, the ideal $I_n$ with basis $(y_k:k\ge n)$, and all subspaces containing the derived subalgebra $I_1$. Hence the intersection of any two nonzero ideals is nonzero, which is stronger than semiprime: every nonzero ideal is essential. But the intersection of all nonzero (=essential) ideals is reduced to zero, which is not essential.