Classify all finitely generated modules $M$ over $R=\mathbb{Z}[i]$ where $5M=0$
By the prime decomposition theorem, I know that $M\cong R^n\oplus(\bigoplus_{i=1,...,u} R/(p_i^{k_i}))$ where $p_i$ are primes in $R$. Since $5M=0$, the torsion-free part is trivial. Then $n=0$. But then I realized that there are infinitely such module $M$ because any finite direct sum of $R/(1+2i)$, $R/(1-2i)$ should work, as 1+2i and 1-2i are irreducible factors of $5$. Is it supposed to have infinite possibilities? Did I do anything wrong?