matrix rings

Last updated January 27, 2022

Properties

1. ( Skolem-Noether theorem) All automorphisms of $M_n(K)$ are inner, i.e. given by $M\mapsto CMC^{-1}$ for some invertible $C$.
2. $\text{Aut}(M_n(K))\cong PGL_n(K)$.

Proofs 2 Consider the homomorphism

\begin{gather} GL_n(K)\to\text{Aut}(M_n(K))\
M\mapsto CMC^{-1}. \end{gather} By (1), this is surjective. We now consider the kernel. Suppose $CMC^{-1}=1$. Since we can make no assumptions on $C$ as it may vary with $M$, we conclude that $M$ is in the center of $GL_n(K)$, hence a scalar matrix. These nonzero scalar matrices are the center $Z$. By definition, $PGL_n(K)=GL_n(K)/Z$, whence the statement.