My professor gave us the following definition:
Let $R$ be a commutative ring. $R$ has the ascending chain condition $(ACC)$ if, for every ascending chain of ideals $$I_{1} \subset I_{2} \subset I_{3} \subset ...$$ in $R,$ there is an $N > 0$ so that $I_{n} = I_{n+1}$ for $n \geq N.$
My question is:
Does that condition means they are equal up to associates? i.e. $I_{n} = u I_{n+1}$ for some unit $u \in R^{\times}$
If that is correct, can anyone show me a rigor proof to that please?
What the equation $I_n = I_{n+1}$ means, i.e. what it is defined to mean, is equality of sets; in other words, it means that $x \in I_n$ if and only if $x \in I_{n+1}$.
On the other hand, if $u$ is a unit, then you should be able to check that $I_{n+1} = u I_{n+1}$, which means that $x \in I_{n+1}$ if and only if $x \in u I_{n+1}$. (Hint on the proof: $x = u (u^{-1} x)$).
Therefore, $I_n = I_{n+1}$ is equivalent to $I_n = uI_{n+1}$ for some unit $u \in R^{\times}$.