Given a ring $A$, and an $A$-module $M$. Let $x_1, . . . , x_n ∈ M$. Say $\operatorname{rank}(M) = n$ if for all $m ∈ M$ there exist unique elements $a_i ∈ A$, $i = 1, . . . , n,$ such that $m=\sum_{i=1}^n a_i x_i$.
Give examples such that
1. $M$ is free of rank $1$ and has a submodule $M_0$ which cannot be generated by less than two elements, and
2. $M$ is finitely generated, but $M_0$ is not finitely generated.
I am stuck on this examples but I am unable to find one. I know that if $A$ is a field then its impossible since M is vector space. I tried to test with $A = \mathbb{Z}$ but it doesn't work for me. Can anyone give me some hints to work on? Thank you in advance!
Set $A=\Bbb Z[x,y]/(xy)$ with $M=A$ and $M_0=(x,y)$.
For an infinite example, instead use $A= \Bbb Z[x_1,x_2,\ldots]/I$ where $I$ is the ideal generated by all cross-products $x_ix_j, i\neq j$. Similarly use $M=A$ and $M_0=(x_1,x_2,\ldots)$.