Let $k$ be a field and $A \supset k$ a ring that is finite dimensional as a $k$-vector space, prove that A is Noetherian and Artinian.
For the first part, I've been trying to use a.a.c condition, so for all chain of ideals of $A$
$$I_1 \subseteq I_2 \subseteq...\subseteq I_k\subseteq...$$ we have that $$V_1 \subseteq V_2 \subseteq...\subseteq V_k\subseteq...$$ where $V_j=A/I_j$ k-vector space of dimension $n_j$. Using that $dim(A)$ is finite, we conclude $$0 \leq n_1 \leq...\leq n_k\leq...\leq dim(A)$$ so necessary $\exists n\in \mathbb{N}: n_n=n_{n+1}=...$ (Suppossing not we have a contradiction). So A is Noetherian.
For Artinian I've made the same but with d.c.c and with the inequality $$0\leq... \leq n_k \leq...\leq n_1\leq dim(A)$$
It's my solution right?
Thank you for your help.
Yes, that is the right idea. Right and left ideals are finite dimensional subspaces of a finite dimensional space, and this prevents them from strictly ascending or descending forever.