Integral Domains: Homework Statement & Equations

[mr.tea]

Homework Statement

Let $R$ be a principal ideal domain and suppose $I_1,I_2,...$ are ideals of $R$ with
$I_1 \subseteq I_2 \subseteq I_3 \subseteq ...$
The Question has two parts: 1. to show that $\cup _{i=0}^{\infty}I_i$ is an ideal.
2. to show that any ascending as above must stabilize, i.e. there is a positive integer $n$ with $I_n=I_{n+1}=...$

Homework Equations

The Attempt at a Solution

My problem is with the second question. I tried to assume for contradiction that for every positive integer $n$, we have $I_n \subsetneq I_{n+1}$ which mean that there is a number $x\in I_{n+1}$ which is not in $I_n$. Since we are in a PID, we can write $I_n = (d), \quad I_{n+1}=(e)$ ( where $d,e$ are the generators). I also got that $d \nmid x$, and I tried to write $\gcd(x,d)$ as a linear combination of them... I have ran out of ideas...

Any hint will be helpful!
Thank you.

 

[fresh_42]

Doesn't part one help you out here?

 

[mr.tea]

Doesn't part one help you out here?

Unfortunately, not too much. The closest relationship between $x$ and $d$ that I have is $x(1-s)=r\cdot d\cdot n$, where $r,s$ came from $\gcd(x,d)=sx+rd$, and $n$ came from $\gcd(x,d) \cdot n=x$

Is it possible that it is related to the fact that we have unique factorization in PID?

 

[fresh_42]

Is it possible that it is related to the fact that we have unique factorization in PID?

If you can use this result, then it's the step in the right direction.