# Artinian Ring => Noetherian: Why is this “proof” wrong?

I'm supposed to show that Artinian rings are Noetherian and my first idea was to take an ascending chain of ideals $I_0 \leq R$:

$I_0 \subsetneq I_1 \subsetneq \cdots$

Taking quotients of $R$ we get a descending chain of ideals:

$R/I_0 \supsetneq R/I_1 \supsetneq \cdots$

which we cannot have as R is Artinian.

I strongly suspect the argument is invalid as analogous reasoning would be able to reverse the implication. But I can't see where to find my presumably elementary mistake. Help will be appreciated!

• The quotients $R/I_i$ aren't ideals of $R$. In particular, they aren't even subsets of $R$, since each one consists of equivalence classes of elements of $R$. – Ravi Fernando Oct 13 '15 at 23:12
• That was it - if you promote your comment to an actual answer I'll flag it as the solution. Thank you! – dom_miketa Oct 13 '15 at 23:13

The quotients $R/I_i$ aren't ideals of $R$. In particular, they aren't even subsets of $R$, since each one consists of equivalence classes of elements of $R$.