Neither Artinian nor Noetherian rings I read a message about the ring of complex entire functions, that is neither Artinian nor Noetherian (see here).

Can you show me other examples of rings that are neither Artinian nor Noetherian?

 A: For any field $k$ consider the ring of polynomials in infnite many variables $k[\{x_n:n\in \mathbb N\}]$. It is not noetherian and hence is not artinian.
A: For commutative rings (edit: I required commutative rings to have a $1$), the condition that $R$ is Artinian is equivalent to the condition that $R$ is Noetherian and has Krull dimension $0$. Thus any commutative ring which is not Noetherian is not Artinian either. One common example is the ring $A$ of algebraic integers. It has an infinite ascending chain of ideals as follows:
$$(2)\subset (\sqrt{2})\subset (\sqrt[3]{2})\subset\cdots$$
and an infinite descending chain of ideals:
$$(2)\supset (2^2)\supset(2^3)\supset\cdots$$
Another example is the ring $R$ of polynomials over the field $k$ in infinitely many variables, which has an infinite ascending chain of ideals:
$$(x_1)\subset(x_1,x_2)\subset(x_1,x_2,x_3)\subset\cdots$$
and an infinite descending chain chain of ideals:
$$(x_1)\supset(x_1^2)\supset(x_1^3)\supset\cdots$$
Note that these are just examples of infinite ascending and descending chains of ideals in these rings; many more exist.
