Searching on this site and others leads to lots of dicussion about localisation at multiplicatively closed subsets of the form $\{f_i^j\}_{j=1}^\infty$ where $\{f_i\}_{i=1}^n$ generate the whole ring R over which we are considering modules.
However I can't find a clear/simple answer about the statement:
$M$ finitely generated as an R-module $\Leftrightarrow$ $M_{\mathfrak{m}}$ is finitely generated for all maximal ideals.
Could anybody help me either proving this or giving a counter example?
Consider $R=\Bbb Z$ and $M=\bigoplus_p \Bbb Z/p\Bbb Z $.