Is this algebra semisimple?

Question

Is this algebra, A, semisimple? And what are its simple modules?

$$\begin{bmatrix}a&b&0\\c&d&0\\0&0&e\end{bmatrix} \subset M_3(k),$$ where $k$ is a field and $a,b,c,d,e$ are elements in $k$.

I know every non $0$ A-module is semisimple, so does it suffice to show that this is an A module? I'm very confused.

Closely related: Why $R$ is semisimple ring iff every $R$-module is semisimple? — hardmath, Mar 20 '19 at 18:40
Thank you, this makes sense. How do I do about finding the simple modules? — Beryl1934, Mar 23 '19 at 12:39

score 0 · Answer 1 · answered Mar 21 '19 at 01:12

0

There're two well-known Theorems

2.A Finite dimensional semisimple algebra over a field k is a direct sum of matrix $M_{n_i}(D_i)$, where D is a division ring over k.

answered Mar 21 '19 at 01:12

abvdd

Thank you everyone. How do I go about finding the simple modules? – Beryl1934 Mar 23 '19 at 12:40

1 Answers1