$R$ a local ring, also a PID. $I,J$ ideals from $R$. Show that $I \subseteq J$ or $J \subseteq I$

Question

$R$ a local ring, also a PID. $I,J$ ideals from $R$. Show that $I \subseteq J$ or $J \subseteq I$

My brief attempt to try use Bezout theorem at a PID. but unsuccess..

Thanks any help.

Answer 1

If it is not a field, it has exactly one prime element (up to associates, of course.) (explain why)

It is a UFD, and apparently everything that is nonzero has a factorization which is a power of p times a unit.

From this, observe the nontrivial ideals are just $(p^i)$ for integers $i>0$.

Answer 2

$I, J$ are each generated by a single element, say $x, y$ respectively. Every ring ideal is contained in a maximal ideal, in this case the (principal) maximal ideal, say, $(t)$. If $x$ is a unit, then $I=R$, so $J\subseteq I$. Similarly if $y$ is a unit. If not, then $x=ut^j$, $y=vt^k$ for some units $u,v$ and $j,k\in\mathbb{N}$. If $j\leq k$, then $y=vt^k=vu^{-1}t^{k-j}x\in I$, so $J\subseteq I$; similarly if $k\leq j$.