I'm trying to solve this problem from my abstract algebra course:
Given $A$ a commutative unital ring. Prove that the following two properties are equivalent:
- Every ideal of $A$ is finitely generated.
- There is no family $\{I_n\}^\infty_{n=1}$ of ideals from $A$ such that $I_n\subsetneq I_{n+1}$ for all $n\geq 1$
I am confused from the second statement, it's hard to imagine for me what does that exactly mean for me to use it in my proof. I started trying to assume every ideal in $A$ can be expressed as $I=(a_1,\dots,a_n)$ for some elements $a_i\in A$ (to be said, it's finitely generated), but I don't know how to get to the other statement from that.
For the reverse implication, I understand in my head that assuming that we have such a family of sets, in order for them to verify $I_n\subsetneq I_{n+1}$ for all $n\geq 1$, they must be always growing in size because they can't be equal, so that leads to an infinite amount of generators, but I don't know how that must be written correctly in a mathematic way.
How can I approach this? Any hint, answer or just help will be appreciated, thanks in advance.