1) Is there any characterization of $\Bbb{Z}$-modules with finite uniform dimensions?
2) Find two $\Bbb{Z}$-modules $N, M$ such that $N\leq M$ and $M\hookrightarrow N$ but $N$ is not isomorphic to $M$?
1) Is there any characterization of $\Bbb{Z}$-modules with finite uniform dimensions?
2) Find two $\Bbb{Z}$-modules $N, M$ such that $N\leq M$ and $M\hookrightarrow N$ but $N$ is not isomorphic to $M$?
Here is an answer to Question 2):
Set $\newcommand{\zz}{\mathbb{Z}}$
$$ N = \bigoplus_{n=2}^{\infty} ( \zz / n\zz ), \qquad
M = \bigoplus_{n=2}^{\infty} ( \zz/ 4n\zz ). $$
Embed $N$ into $M$ by embedding each summand $\zz/n\zz$ into $\zz/4n\zz$.
Embed $M$ into $N$ by sending $\zz/4n\zz$ to the summand $\zz/4n\zz$ at positions $4$, $8$, $\dots$ in $N$. The modules $M$ and $N$ can not be isomorphic because $N$ contains an $x$ with $2x=0$ and $x \notin 2N$, but $M$ does not contain such an element $x$.