an R-submodule of is in the form where is an ideal of containing so for some and thus

for some and units it's clear now that if then and if then it's now obvious that is indecomposable because if for some submoules

then we'll have but, as we just showed either or so we must have either or

suppose has at least two distinct prime factors. let be a prime factor of so we have where therefore see that and so(ii) Let R be a PID and d is non_zero, non_unit. If R/(d) is indecomposable, show that d~p^n for some prime p in R.

Can you give me some hints how to do this question please?

Thank you so much

which contradicts indecomposablity of thus is the only prime factor of and hence and are associates.