How to prove that the ring of upper trianglular matrices is not semisimple?

Question

I was wondering how to prove that the ring $$R= \bigg\{ \ \left( {\begin{array}{cc} a & b \\ 0 & c \\ \end{array} } \right): \ a,b,c \in \mathbb{C} \ \bigg\} $$ is not semisimple. One way that i followed is considering the exact sequence $$ 0 \to U \to V \to V/U \to 0 $$ where $$U=\bigg\{ \ \left( {\begin{array}{cc} a \\ 0 \\ \end{array} } \right): \ a \in \mathbb{C} \ \bigg\} \text{ and } V= \bigg\{ \ \left( {\begin{array}{cc} a \\ b \\ \end{array} } \right): \ a,b \in \mathbb{C} \ \bigg\} $$ with purpose to prove that is not splitting. In this case, i can't find any function $f: V/U \to V$ or an other argument to get the result.

Answer 1

That's fine.

For another thing, $\left\{\begin{bmatrix}0&b\\0&0\end{bmatrix}\middle|\,b\in\mathbb C\right\}$ is a nonzero nilpotent ideal.

Yet another way: because of the Artin-Wedderburn theorem, the only possible $3$ dimensional semisimple $\mathbb C$ algebra is $\mathbb C\times\mathbb C\times \mathbb C$, which is commutative, and this isn't.