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A ring R is said to satisfy the ascending chain condition (ACC) if for every set of ideals, $I_1 \subset I_2 \subset I_3 ...$ there exists a natural number N such that $I_n = I_N$ for all $n \ge N$.

We know that for integers which are PID, $\gcd(n, m) = an + bm$ where $a,b \in \mathbb{Z}$. So if $m$ and $n$ are prime, the ideal $\langle n,m\rangle = R$ since it would contain $1$. So the principal ideal generated by prime integers are subset of the ring but neither $\langle 2\rangle$ nor $\langle 3\rangle$ are subset of one another. So how could integers be ACC if one principal ideal cannot be contained by another?

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  • $\begingroup$ What do you think ACC stands for? $\endgroup$ Commented Sep 16, 2022 at 0:13
  • $\begingroup$ Ascending Chain condition. I've edited the question to include its definition $\endgroup$ Commented Sep 16, 2022 at 0:20
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    $\begingroup$ ACC says ascending chains terminate. It doesn't say there aren't incomparable ideals. $\endgroup$ Commented Sep 16, 2022 at 0:24
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    $\begingroup$ I see, you seem to be under the impression that every set the ideals have to satisfy $I_1 \subseteq I_2 \subseteq I_3 \subseteq \cdots$. That's not the case. It says that IF $I_1 \subseteq I_2 \subseteq I_3 \subseteq \cdots$ THEN exists N such that $I_n = I_N$ for all $n>N$ $\endgroup$ Commented Sep 16, 2022 at 0:25

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ACC is a property of a partially ordered set, which is not necessarily a totally ordered set. Specifically, a set $S$ with a partial order $\leq$ is said to satisfy if ACC if for any sequence $(x_1, x_2, \dots)$ with $x_1 \leq x_2 \leq \dots$, there exists $n \in \mathbb{N}$ such that $x_n = x_{n + 1} = \dots$.

So in particular case of $\mathbb{Z}$ considered as a ring and $S$ the set of ideals of $\mathbb{Z}$ partially ordered by inclusion, the fact that the ideal $(2)$ does not contain the $(3)$ or vice-versa is irrelevant to determining if $\mathbb{Z}$ satisfies ACC.

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