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.


1 Answer 1


There're two well-known Theorems

  1. A ring is semisimple iff all its modules are semisimple.

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.

  • $\begingroup$ Thank you everyone. How do I go about finding the simple modules? $\endgroup$
    – Beryl1934
    Commented Mar 23, 2019 at 12:40

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