- $\mathcal Z(R_R)\subseteq \ell.ann(soc(R_R))$, and they are equal when $R$ is right Artinian.
- $J(R)=\mathcal Z(R_R)$ for principally injective rings
- $Nil_\ast(R)\subseteq Nil^\ast(R)\subseteq J(R)$
- $soc(_RR)=soc(R_R)$ in dual rings
- 2-primal implies Dedekind finite
- A right simple-injective, right Kasch ring is right dual
- Akizuki–Hopkins–Levitzki
- Amitsur's nil radical theorem
- Amitsur-Levitzki theorem
- Andrunakievič's theorem on subdirect representations of reduced rings
- Artin - Wedderburn theorem
- Artinian, Noetherian, semisimple, semiprimitive, primitive are Morita invariant
- Baker–Heegner–Stark theorem on $\mathbb Q[\sqrt{d}]$
- Camillo's theorem on semihereditary polynomial rings
- Characterization of self-injective Leavitt path algebras
- 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
- Classical quotient ring of a commutative reduced ring
- Commutative equivalents to von Neumann regularity
- Conditions equivalent to von Neumann regularity in self-injective rings
- Evans' Cancellation Theorem
- Every countable dimensional $F$ algebra embeds in $RCFM_\omega(F)$
- Finite generation of projective ideals
- Formanek-Snider primitive group-ring theorem
- fp-injectivity in serial rings
- Frobenius' theorem
- Goldie's theorem on semisimple classical quotient rings
- Inheritance of UFD to power series rings
- Jacobson density theorem
- Kaplansky's theorem on modules of right hereditary rings
- Kaplansky's theorem on PI algebras
- Kaplansky's theorem on projective modules over a local ring
- Kaplansky, Jaffard, Ohm theorem for constructing Bézout domains
- Krull dimension of an infinite product
- Krull dimension of an infinite product of zero dimensional rings
- Krull's valuation domain theorem
- Lawrence's theorem on self-injective rings
- Levy's characterization of almost-self-injective Noetherian rings
- Maschke's Theorem (on when a group algebra is semisimple.)
- Nilpotency of $\mathcal Z(R_R)$
- OBVIOUS
- Ore's theorem on the classical quotient ring of a domain
- Semicommutative rings from Armendariz rings
- Socles coincide in semiprime rings
- Structure of Noetherian distributive rings
- Structure of self-injective von Neumann regular rings
- Uniserial Noetherian rings
- Unit-regular rings are clean
- Wedderburn's little theorem
- 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