Is it true that if a module $M$ has finite uniform dimension then the same is true for its homomorphic images ?



On modules with finite uniform dimension
Here is a (very easy) counterexample. 


The rationals $\mathbb{Q}$ form a uniform module over $\mathbb{Z}$, and thus $\mathbb{Q}$ has uniform dimension $1$, but
$\mathbb{Q}/\mathbb{Z}$ is an infinite direct sum of its $p$-torsion parts and thus has infinite uniform dimension.


