Examples of non Noetherian rings satisfying ACC condition for annihilators

Question

I am reading Goddearl and Warfield book and I see that the common examples of Goldie rings are Noetherian semiprime rings. However I am wondering about rings that satisfy the ascending chain condition on annihilators which are not right Noetherian rings. Does anyone know some example?

Answer 1

Obviously any non-Noetherian domain works:

1. $\mathbb R[x_1,x_2,\ldots]$ in countably many variables.
2. $\mathbb Z + x\mathbb Q[x]\subseteq \mathbb Q[x]$ is another example that doesn't even have the ACC on principal ideals.
3. The ring of holomorphic functions on $\mathbb C$ is another example.

Here are some rings which are not left Noetherian but have the ACC on left annihilators. You may also find something different in some rings which are not left Noetherian but have the ACC on right annihilators.

If you were really wondering about non-right-Noetherian right-Goldie rings, then the three examples above still qualify. There are actually more hits for non-right Noetherian, left Goldie rings, in particular $\mathbb Z\langle x,y\rangle/\langle y^2, yx\rangle$.