I have learned a fact that the matrix ring $M_n(D)$ over a division ring $D$ is both Artinian and Noetherian.
Now let $R$ be an arbitrary non-commutative division ring (with $1$).
I am thinking if R is Noetherian. My book doesn't mention this. I guess the answer is negative but I cannot come up with a counter-example.
I know $R$ is Noetherian if and only if every left ideal and right ideal of $R$ are finitely generated. So, equivalently, I am seeking for a non-commutative division ring which has some infinitely generated left (or right) ideals.
Can anyone give me such a counter-example? Thanks for help.
The left/right ideals of a division ring $D$ are $(0)$ and $D$: if $I$ is an nonzero left ideal, pick $x\in I\setminus\{0\}$. By assumption on $D$, $x$ is invertible. Thus, for all $d\in D$, we have $d=(dx^{-1})x\in I$, so $I=D$.
A similar argument applies for right ideals. Now $(0)$ and $D$ are finitely generated, since they are generated by $0$ and $1$ respectively. Hence $D$ is noetherian.