Badly behaved, but easy-to-manipulate examples of rings to test hypotheses on

Question

In calculating examples in mathematics it's often useful to have a quite misbehaving but easy-to-manipulate object to test hypotheses on. Examples are the function $ f(x)=\begin{cases} 0 & \text{ if } x \in \Bbb{Q}\\ 1& \text{ if } x\not \in \Bbb{Q} \end{cases}$ in analysis, or the Baumslag-Solitar groups $B(n,m)$ in group theory.

Do there exist rings that are like this? If so, which are your favourites?

At the moment I tend to use $\Bbb{Z}$ or $\mathbb{Z}/d \Bbb{Z}$, which is really sub-optimal: it is nearly impossible to tell whether a property one finds is specific to the ring I'm using or not. I especially find this troublesome when investigating properties of ideals.

Answer 1

Here are some example from complex analysis.

The ring of holomorphic function $\mathcal O(\mathbb C)$ has lot of interesting properties, for example it is a Bezout ring but not a PID.

Another examples are here, can you try to find the noetherian one ? (this is an exercise in Atiyah and Mcdonald's Introduction to Commutative Algebra)

• The rational functions with no poles on the circle $|z|=1$
• The power series with a positive radius of convergence
• The power serie with an infinite radius of convergence
• The polynomials in $z$ with first $k$ derivatives which vanishes at $z=0$
• The ring of polynomial in $z,w$ which first $k$ derivatives with respect to $w$ vanishes for $z=0$.

If you want to find many examples of rings, you can look in algebraic geometry or number theory where rings and ideals arise in a very natural way.