- Baker–Heegner–Stark theorem on $\mathbb Q[\sqrt{d}]$
- Camillo's theorem on semihereditary polynomial rings
- Characterization of self-injective Leavitt path algebras
- Formanek-Snider primitive group-ring theorem
- Inheritance of UFD to power series rings
- Kaplansky, Jaffard, Ohm theorem for constructing Bézout domains
- Krull's valuation domain theorem
- Maschke's Theorem (on when a group algebra is semisimple.)
- Semicommutative rings from Armendariz rings
- When a group ring is Artinian
- When a group ring is hereditary (Dicks's theorem)
- When a group ring is local
- When a group ring is Noetherian
- When a group ring is perfect
- When a group ring is prime
- When a group ring is principally injective
- When a group ring is self-injective
- When a group ring is semilocal
- When a group ring is semiprime
- When a group ring is von Neumann regular
- When skew polynomial rings are right duo