An $A$-module $M$ is locally cyclic if every submodule of $M$ of finite type (finitely generated) is cyclic.
(i) Show that every submodule of a locally cyclic module is locally cyclic.
(ii) Prove that if $M$ is locally cyclic and if $f:M \to N$ is a modules epimorphism, then $N$ is locally cyclic.
I'll write what I've done up to now:
For (i):
Let $N$ be a submodule of $M$ of finite type. Then, by hypothesis $N=<n>$. Now let $N'$ be a submodule of $N$ of finite type, but then $N'$ is also a submodule of $M$ so $N'$ is cyclic.
For (ii):
Let $N'$ be a finitely generated submodule of $N$. Then $N'=<y_1,...,y_n>$, by hypothesis, there exist $x_1,...,x_n \in M$ such that $f(x_i)=y_i$ for all $1 \leq i \leq n$. We consider $M'=<x_1,...,x_n>$, since $M'$ is finitely generated, then $M'=<x>$ for some $x \in M$. I am going to prove $N'=<f(x)>$. Take $y \in N'$, then $y=\sum_{i=1}^n a_iy_i=\sum_{i=1}^n a_if(x_i)=f(\sum_{i=1}^na_ix_i)$. Since $\sum_{i=1}^n a_ix_i \in M'=<x>$, we have $\sum_{i=1}^n a_ix_i=ax$ for some $a \in A$., but then $y=f(ax)=af(x)$, from here it follows $N'=<f(x)>$.
I am having some doubts with my proof in (i) since I am not using the fact that $N$ is cyclic, I would appreciate if someone could take a look at the two parts of the problem and correct me if I've made any mistakes. Thanks in advance.