At the end of Farb and Dennis 'Noncommutative Algebra', page 213, is an exercise. It specifies thirteen different rings, such as $\Bbb Z$, $\Bbb Z/n\Bbb Z$, $\Bbb{C}[x]$, etc.
It then asks you to do three different chains of things:
- Determine if each ring is simple, semisimple, radical-free, artinian, noetherian, primitive, semi-primitive, prime, von-Neumann regular,
- Compute $Z(R), J(R), R^\times$ the zero divisors of $R$, the nilpotents of $R$, the idempotents of $R$,
- Classify finitely generated $R$-modules which are simple, semi-simple, of finite-length, free, projective, injective.
I think this exercise is excellent. Is there a similar exercise anywhere, in particular, say, for homological algebra. Perhaps one would specify a bunch of rings, and ask for you to determine projective dimension, injective dimension, etc? If there is no such list, what would be a good list of homological properties to consider, and a good list of rings to consider, so we can construct our own analogous exercise.
If it is just commutative rings, that is fine too.