= Solution
{c}
\b[A <ring> is a <Noetherian ring> if it satisfies the <ascending chain condition> on <ideals>]: every chain
$$
I_1\subseteq I_2\subseteq I_3\subseteq\cdots
$$
eventually becomes constant. Equivalently, every <ideal> has a finite <generating set of an ideal>. To see this equivalence, if an <ideal> were not finitely generated, one could repeatedly choose an element outside the <ideal> generated by the preceding choices, producing a strictly ascending chain. Conversely, the union of an ascending chain is an <ideal>; if it is generated by finitely many elements, all of them lie in one term of the chain, and every later term equals that term.
\b[A <ring> is an <Artinian ring> if it satisfies the <descending chain condition> on <ideals>]: every chain
$$
I_1\supseteq I_2\supseteq I_3\supseteq\cdots
$$
eventually becomes constant. These are stabilization conditions on all <ideals>, not merely on <principal ideals>.
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