- Akizuki–Hopkins–Levitzki
- Artinian, Noetherian, semisimple, semiprimitive, primitive are Morita invariant
- Characterizations of hereditary rings
- Characterizations of quasi-Frobenius rings
- Characterizations of right nonsingular rings
- Characterizations of right perfect rings
- Characterizations of self-injective rings
- Characterizations of semiperfect rings
- Characterizations of semisimplicity
- Characterizations of von Neumann regular rings
- Chase's theorem on products of flat modules
- Evans' Cancellation Theorem
- Kaplansky's theorem on modules of right hereditary rings
- Kaplansky's theorem on projective modules over a local ring