This question came up when I was working on the following problem
Let $R$ be an integral domain. Prove that if the following two conditions hold then $R$ is a Principal Ideal Domain:
i) any two nonzero elements $a$ and $b$ in $R$ have a greatest common divisor which can be written in the form $ra+sb$ for some $r,s \in R$.
ii) if $a_1, a_2, a_3,\ldots$ are nonzero elements in $R$ such that $a_{i+1}|a_i$ for all $i$, then there is a positive integer $N$ such that $a_n$ is a unit times $a_N$ for all $n \geq N$.
I have managed to prove this by assuming that an arbitrary ideal in an integral domain $R$ can be written as $I = (x_1, x_2, \ldots)$ for some $x_i \in R$, if however there existed an ideal that could only be generated by an uncountable set in $R$ then my proof would not work.
The obvious counterexample seems to work: that is, $F[\{x_i\mid i\in I\}]$ for an uncountable index set $I$ and field $F$. The ideal generated by the $x_i$ is not countable generated.
It’s hard to imagine how your argument relied on countable generation. Could you explain more clearly what you argued? Otherwise it is hard to help.