I was asked to prove that every finite, commutative ring is Noetherian.
My attempt:
Let $R$ be a finite ring. Let $I_1\subseteq I_2\subseteq I_3....$ be a chain of ideals of $R$. Since $R$ is finite, with $K$ member, the number of ideals on $R$ is finite, at most $2^K$. Consider the two cases:
(1) For all $i\neq j$ , $I_i\neq I_j$. Therefore, the chain must stabilise and become constant, for otherwise, we would have more than $2^K$ ideals, absurd.
(2) For all $i\neq j$, $I_i= I_j$. This is trivial.
Note, (1) automatically implies that if there exists $i\neq j$, then $I_i\neq I_j$. Hence we are done.
Is the logic correct?