define where is a cyclic group of prime order p, then R is a module over itself (it's a ring). If then I need to show that the set is an indecomposable module.
let note that is the augmentation ideal of and i'll show that is a minimal ideal of which solves the problem. it's clear that
now since we only need to prove that if is an ideal of such that then either or this is very easy to see:
since is a PID, we have for some since we have for some but which means for
some thus and hence but since is irreducible over we must have either or thus
either or