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!