Theorems in ring theory

• All theorems
• Connections with modules
• Properties of constructions
• Properties of special subsets
• Structure/splitting theorems
• (Uncategorized)
• Characterization of properties
• Connections with rings of quotients

• OBVIOUS
• Conditions equivalent to von Neumann regularity in self-injective rings
• Maschke's Theorem (on when a group algebra is semisimple.)
• Artin - Wedderburn theorem
• Structure of Noetherian distributive rings
• When a group ring is self-injective
• Akizuki–Hopkins–Levitzki
• Amitsur's nil radical theorem
• When a group ring is principally injective
• Uniserial Noetherian rings
• When a group ring is semilocal
• When a group ring is local
• When a group ring is perfect
• $J(R)=\mathcal Z(R_R)$ for principally injective rings
• Kaplansky, Jaffard, Ohm theorem for constructing Bézout domains
• Krull's valuation domain theorem
• fp-injectivity in serial rings
• Amitsur-Levitzki theorem
• Unit-regular rings are clean
• Evans' Cancellation Theorem
• Lawrence's theorem on self-injective rings
• A right simple-injective, right Kasch ring is right dual
• Artinian, Noetherian, semisimple, semiprimitive, primitive are Morita invariant
• 2-primal implies Dedekind finite
• Wedderburn's little theorem
• Frobenius' theorem
• Jacobson density theorem
• Chase's theorem on products of flat modules
• Semicommutative rings from Armendariz rings
• $soc(_RR)=soc(R_R)$ in dual rings
• $\mathcal Z(R_R)\subseteq \ell.ann(soc(R_R))$, and they are equal when $R$ is right Artinian.
• Nilpotency of $\mathcal Z(R_R)$
• $Nil_\ast(R)\subseteq Nil^\ast(R)\subseteq J(R)$
• Structure of self-injective von Neumann regular rings
• Commutative equivalents to von Neumann regularity
• Baker–Heegner–Stark theorem on $\mathbb Q[\sqrt{d}]$
• Characterizations of quasi-Frobenius rings
• Characterizations of von Neumann regular rings
• Characterizations of hereditary rings
• Characterizations of semisimplicity
• Characterizations of semiperfect rings
• Characterizations of right nonsingular rings
• Characterizations of right perfect rings
• Goldie's theorem on semisimple classical quotient rings
• Ore's theorem on the classical quotient ring of a domain
• Classical quotient ring of a commutative reduced ring
• Kaplansky's theorem on PI algebras
• Camillo's theorem on semihereditary polynomial rings