# Every principal ideal domain satisfies ACCP.

Every principal ideal domain $$D$$ satisfies ACCP (ascending chain condition on principal ideals)

Proof. Let $$(a_1) ⊆ (a_2) ⊆ (a_3) ⊆ · · ·$$ be a chain of principal ideals in $$D$$. It can be easily verified that $$I = \displaystyle{∪_{i∈N} (a_i)}$$ is an ideal of $$D$$. Since $$D$$ is a PID, there exists an element $$a ∈ D$$ such that $$I = (a)$$. Hence, $$a ∈ (a_n)$$ for some positive integer $$n$$. Then $$I ⊆ (a_n) ⊆ I$$. Therefore, $$I = a_n$$. For $$t ≥ n$$, $$(a_t) ⊆ I = (a_n) ⊆ (a_t)$$. Thus, $$(a_n) = (a_t)$$ for all $$t ≥ n$$.

I have prove $$I$$ is an ideal in the following way:-

Let $$x,y\in I$$. Then there exist $$i,j \in \mathbb{N}$$ s.t. $$x \in (a_i)$$ & $$y \in (a_j)$$.
Let $$k \in \mathbb{N}$$ s.t $$k>i,j$$.
Then $$x \in (a_k)$$ & $$y \in (a_k)$$.
as $$(a_k)$$ is an ideal $$x-y \in (a_k)\subset I$$ and $$rx,xr \in (a_k)\subset I$$.
So $$I$$ is an ideal.

Is it correct?

## 2 Answers

Your proof is right but you can let t = max(i,j) and any k > t.

Yes. Said more simply $\ (a_1) \subseteq (a_2) \subseteq \cdots\subseteq (a_1,a_2,a_3,...)\stackrel{\rm PID} = (c_1 a_1 +\cdots + c_k a_k) \subseteq (a_k)$

• It would not hurt to turn your decorated equal sign into an actual explanation of what you mean, as it is quite not transparent... Once you do that, I would be surprised to find anything different from what the OP wrote! – Mariano Suárez-Álvarez Jun 2 '13 at 17:36
• The key idea of the proof is hidden in the annotation of the equal sign, rather... – Mariano Suárez-Álvarez Jun 2 '13 at 17:47
• @Mariano The equal sign denotes an equality of ideals, deduce by the PID hypothesis. What is not transparent about that? – Key Ideas Jun 2 '13 at 17:52
• I don't think it is constructive to continue this. As I said above, I do not find your «more simple proof» to be a complete argument: explaining what exactly the annotation means requires something equivalent to what the OP wrote (in a context where the OP is actually asking why exactly is the union of his ideals an ideal!). At best, I find what you wrote as a stenograph for a complete argument. You apparently do not agree, but that is irrelevant to my point. – Mariano Suárez-Álvarez Jun 2 '13 at 17:55
• @Mariano That's precisely what it is intended to be, a one-line summary of the key idea of the proof (as are many of my hints). Note: I meant to write "said more simply...". I've added the "said" to avoid any further confusion. – Key Ideas Jun 2 '13 at 17:58