TITLE: Left noetherian ring but not right noetherian ring
QUESTION [6 upvotes]: I am starting to learn about noetherian ring. Actually, I see that many books mention of right noetherian but not left noetherian ring.
I would like to find some example of left noetherian but not right noetherian ring.
I tried to work on the set of $2\times 2$ - matrices but not effective.
I also think about the case that "Left noetherian implies right noetherian" but there is no clue to prove.
Can you please give me a hint?
Thank you so much.

REPLY [7 votes]: Well, this is exactly what the Database of Ring Theory is supposed to handle. Here is the query. It yields five rings that are one-sided Noetherian:

Quotient of the free algebra $\mathbb Z\langle x,y\rangle$ by the ideal  $(y^2,yx)$
Twisted polynomials $k[x;s]$ for a division ring $k$ with endomorphism s which isn't an automorphism. The twist is given by $xa=xs(a)$.
Triangular ring $\begin{bmatrix}R& R\\ 0& Q\end{bmatrix}$ where $[R:Q]$ is an infinite degree field extension.
Triangular ring $\begin{bmatrix}Q&Q \\ 0&Z\end{bmatrix}$ where $Z$ is a commutative Noetherian domain not equal to its field of fractions $Q$. (This is the one previously mentioned in Joseph Heavner's answer.)
Let $s$ be a field endomorphism from $K$ to $K$ such that the image $L$ has infinite index in $K$. Define multiplication on $R=K\times K$ by $(x,y)(x',y')=(xx',s(x)y'+yx')$. $R$ is the ring. (This one is right Noetherian but not left Noetherian.)