I have to figure out how to show the following ring is not left Noetherian:
$\prod _{1<n\in \mathbb{N}}\mathbb{Z}_{n}$
It is the direct product of all quotient rings.
I have been given the hint that if:
$A_{i+1}\setminus A_{i}\neq \varnothing$
That is, if the elements of an ideal's predecessor are removed from it, you are not given the empty set and that is enough of a basis to show that the ring you've been given is not left Noetherian.
I'm not really sure how that works, so any help given would be greatly appreciated.