Exercise 24, § 6

Let K be a field of arbitrary characteristic $p$. We again adopt Definition 1.

(a) Show that, unless $p = n = 2$, the only non-trivial ideals of $gl(n, K)$ are the 1-dimensional centre $\mathfrak{z}$ with basis $\sum_{i=1}^{n} E_{ii}$ and the Lie algebra $sl(n, K)$ consisting of the matrices of trace 0. (Except in the exceptional case indicated, note that if an ideal $\mathfrak{a}$ contains one of the $E_{ij}$, it necessarily contains $sl(n, K)$. If $\mathfrak{a}$ contains an element not belonging to $\mathfrak{z}$, by multiplying this element by at most four suitably chosen elements $E_{ij}$, a non-zero multiple of one of the $E_{ij}$ is obtained.) If $n$ is not a multiple of $p$, show that $gl(n, K)$ is the direct sum of $\mathfrak{z}$ and $sl(n, K)$ and $sl(n, K)$ is simple. If on the contrary $n$ is a multiple of $p$ and $n > 2$, show that $sl(n, K)/\mathfrak{z}$ is simple (same methods).

(b) Show that, for $n$ a multiple of $p$ and $n > 2$, $gl(n, K)/\mathfrak{z}$ has radical {0}, but admits the quotient $gl(n, K)/sl(n, K)$ which is Abelian.