Left noetherian ring but not right noetherian ring

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.

Solution 1:

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:

1. Quotient of the free algebra $\mathbb Z\langle x,y\rangle$ by the ideal $(y^2,yx)$

2. Twisted polynomials $k[x;s]$ for a division ring $k$ with endomorphism s which isn't an automorphism. The twist is given by $xa=xs(a)$.

3. Triangular ring $\begin{bmatrix}R& R\\ 0& Q\end{bmatrix}$ where $[R:Q]$ is an infinite degree field extension.

4. 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.)

5. 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.)

Solution 2:

The ring $$R = \left\{ \begin{pmatrix} p & q \\ 0 & m \end{pmatrix}: p,q \in \mathbb{Q} \, \text{ and } \, m \in \mathbb{Z} \right\}$$ is an example of a left Noetherian but not right Noetherian ring. You can find a proof of that fact in Section 2.4.1 of these notes.

It is certainly not the case that left Noetherian implies right Noetherian or vice-versa. The two are slightly different concepts depending on whether the chain condition is satisfied by left or right ideals. If a ring is met by both, in other words if it is both left Noetherian and right Noetherian, then we simply call it Noetherian.

Chances are that most examples you will encounter will be either Noetherian or neither left nor right Noetherian. This is because most people work in commutative rings, where a ring is actually left Noetherian if and only if it is right Noetherian, that is all left (respectively, right) Noetherian rings are simply Noetherian rings. The proof of this fact should be clear. (Hint: What can you say about left and right ideals in a commutative ring?)

You probably should not worry too much about the distinction between left and right Noetherian rings, similar to how few need to worry much about pseudo-rings, despite some authors insisting on defining rings in such a way that they need not have an identity. Of course, there are some cases where we need these distinctions, but they are about as conceptually difficult as they are mathematically deep, which is to say not very.