2
$\begingroup$

How can I prove that if $R$ is a finite dimensional algebra over a field then $R$ is simple as a ring if and only if it has a faithful simple left $R$-module?

  • 0
    yeah yeah, it's ok2012-05-06

1 Answers 1

1

OK: if you count Artin-Wedderburn among the simple results you can use, then let's try this.

In one direction, every (unital, nonzero) module over a simple ring with unity is faithful.

In the other direction, the Jacobson radical $rad(R)$ is clearly zero if you look at it as "The intersection of annihilators of simple right $R$-modules." Since an Artinian ring with $rad(R)=\{0\}$ is semisimple, then $R$ is at least semisimple.

Since $R$ is semisimple you can embed your simple module in $R$ as a minimal right ideal $S$. Since nonisomorphic minimal right ideals would annihilate your simple module, there is only one isotype of minimal right ideal. From Artin-Wedderburn theory, $R=\Sigma${minimal right ideals isomorphic to $S$}$\cong M_n(D)$ for a division ring $D$. So, $R$ would be simple.