Can every finitely generated module be written as a sum of simple modules?

Question

Let $R$ be a ring with unity. Take for granted that an $R$-module $M$ is simple if and only if it is cyclic, and that if $m \in M$ then $Rm$ is a submodule of $M$.

Let $M$ be an $R$-module and suppose $N=\{m_1, \ldots ,m_k \}$ generates $M$. Then each $Rm_i$ is a simple module, and since $span(N)=M$, it follows that $$Rm_1 \oplus \ldots \oplus Rm_k=M ,$$ and the result is proven.

Is the above true?

Answer 1

Take for granted that an $R$-module $M$ is simple if and only if it is cyclic,

At this point, you have implicitly assumed $R$ is a division ring, so every $R$ module is a sum of simple modules (copies of $R$).

since $span(N)=M$, it follows that $$Rm_1 \oplus \ldots \oplus R_k=m ,$$

This assumes that your generating set is “a basis” which is true when $R$ is a Division ring if you additionally assume it is a minimal generating set.

So the result is true, but you haven’t established it.