Background
Definition 1: A ring $R$ is said to satisfy the $\textbf{ascending chain condition}$ ($\textbf{ACC}$) for left (right) ideals if for each sequence of left (right) ideals $A_1, A_2, \ldots$ of $R,$ with $A_1\subset A_2\subset\ldots$ there exists a positive integer $n$ (depending on the sequence) such that $A_{n}= A_{n+1}=\ldots$. $R$ is said to satisfy the descending chain condition} (DCC) for left (right) ideals if for each sequence of left (right) ideals $A_1, A_2, \ldots$ of $R,$ with $A_1\supset A_2\supset\ldots$ there exists a positive integer $n$ (depending on the sequence) such that $A_{n}= A_{n+1}=\ldots$.
Definition 2: A ring which satisfies the ACC for left (right) ideals is called a left (right) Noetherian ring.
Definition 3: An integral domain $R$ satisfies the ascending chain condition on principal ideals provided that whenever $(a_1)\subset (a_2)\subset(a_3)\subset\ldots$, then there exists a positive integer $n$ such that $(a_i)=(a_n)$ for all $i\geq n$.
Lemma 1: Every principal ideal domain $R$ satisfies the ascending chain condition on principal ideals.
Questions:
In the above three definitions, and the lemma, i cited the definition of what a Noetherian/Artinian ring is and also that in an integral domain, a principal ideal domain satisfies the ascending chain condition. I tried to find out if given a ring that satisfies the ascending chain condition or descending chain condition, under what circustances can such a ring be made into a principal ideal domain. I know such a ring has to be at first an integral domain. I try looking up at the text Examples of commutative rings by Harry C. Hutchins. There is also the more advanced text Chain condition in Commutative rings by Ali Benhissi. I was not able to find anythign. As a matter of fact, I tried looking up in various algebra text trying to find under what condition can a ring be consider to be a principal ideal domain. But I am not able to find any theorems about it. I am wondering if any of the expert algebraists in the community can shed light on the matter.
Thank you in advance
