If $S$ is simple a module over a ring $R$ which is noetherian, hereditary and every simple module is injective, then $S$ is finitely presentated.

Question

Let $R$ be a left noetherian and left hereditary ring , also suppose every simple left module $M$ over $R$ is injective. Prove that a simple left module $S$ over $R$ is finitely presentated. So Im supposed to find and exact sequence $$\bigoplus_{i=1}^{n} R \to \bigoplus_{i=1}^{m} R \to S \to 0$$ where $n,m \in \mathbb{N}$. Some of my ideas are that as I already know that as $R$ is left hereditary then $p.d(S) \leq 1$ which is somehow why this exact sequence make sense. Also, as $S$ is simple over $R$, then $S$ is cyclic which means it is finitely generated which means $S$ can be covered by copies of $R$, also here I want to use is $R$ Noetherian to get an injection into the direct sum of copies of $R$ which cover $S$. Thanks.

Answer 1

Since $S$ is simple, it is cyclic, and you have a homomorphism $R\to S$.

Since $R$ is Noetherian, the kernel is finitely generated.

This, AFAICT proves $S$ is finitely presented, without any of the other assumptions. You can then get your sequence by mapping onto the kernel with a free module.

In fact, one should also note that every finitely generated left $R$ module over a left Noetherian ring $R$ is automatically finitely presented.