Let us call a (possibly non-commutative) ring $R$ "very good" if every finitely generated torsion left $R$-module is cyclic. Here is an example of such a ring:

Let $k=\mathbb{C}((t))$ and let $R=k[\partial]$ denote the ring of differential polynomials. The addition in this ring is defined as usual, but the multiplication is adjusted by the rule $$ \partial a = a\partial + \frac{d}{dt} a,\quad \quad \forall a\in k.$$

Then it is known that every finitely generated $R$-module is a direct sum of a free module together with a cyclic module. In other words, every finitely generated torsion left $R$-module is cyclic. This fact is known as the cyclic vector theorem and has important consequences for the study of linear differential equations.

Question 1: What other examples of "very good" rings are known?

Question 2: Is there a general abstract property we can put on a ring which guarantees that it is "very good"?

In regards to Question 2, it helps to have the following fact in mind: if we require $R$ to be a PID, then every finitely generated module over $R$ will be a direct sum of cyclic modules. But $\mathbb{Z}$ is an example of a PID which is not "very good"! ($\mathbb{Z}/{2}\oplus \mathbb{Z}/4$ is not cyclic). Thus in some sense, $k[\partial]$ is better than $\mathbb{Z}$. I would like to know if there is a way to formalise this observation using (non-commutative) ring theory.