Ring properties

Here is a list of properties any ring can have.

(Switch to the commutative ring property index for additional properties defined for commutative rings.)

2-primal Abelian ACC annihilator (left) ACC annihilator (right) ACC principal (left) ACC principal (right) Artinian (left) Artinian (right) Baer Bezout domain (left) Bezout domain (right) Bezout (left) Bezout (right) clean cogenerator ring (left) cogenerator ring (right) coherent (left) coherent (right) cohopfian (left) cohopfian (right) commutative connected continuous (left) continuous (right) CS (left) CS (right) DCC annihilator (left) DCC annihilator (right) Dedekind finite distributive (left) distributive (right) division ring domain dual (left) dual (right) duo (left) duo (right) essential socle (left) essential socle (right) exchange FI-injective (left) FI-injective (right) finite finitely cogenerated (left) finitely cogenerated (right) finitely generated socle (left) finitely generated socle (right) finitely pseudo-Frobenius (left) finitely pseudo-Frobenius (right) finite uniform dimension (left) finite uniform dimension (right) free ideal ring (left) free ideal ring (right) Frobenius fully prime fully semiprime Goldie (left) Goldie (right) hereditary (left) hereditary (right) $I_0$ IBN Ikeda-Nakayama (left) Ikeda-Nakayama (right) Kasch (left) Kasch (right) lift/rad local nilpotent radical nil radical NI (nilpotents form an ideal) Noetherian (left) Noetherian (right) nonsingular (left) nonsingular (right) nonzero socle (left) nonzero socle (right) Ore domain (left) Ore domain (right) Ore ring (left) Ore ring (right) orthogonally finite perfect (left) perfect (right) $\pi$-regular polynomial identity primary prime primitive (left) primitive (right) principal ideal domain (left) principal ideal domain (right) principal ideal ring (left) principal ideal ring (right) principally injective (left) principally injective (right) pseudo-Frobenius (left) pseudo-Frobenius (right) quasi-continuous (left) quasi-continuous (right) quasi-duo (left) quasi-duo (right) quasi-Frobenius reduced reversible Rickart (left) Rickart (right) self-injective (left) self-injective (right) semicommutative (SI condition, zero-insertive) semi free ideal ring semihereditary (left) semihereditary (right) semilocal semiperfect semiprimary semiprime semiprimitive semiregular semisimple serial (left) serial (right) simple simple Artinian simple-injective (left) simple-injective (right) simple socle (left) simple socle (right) stable range 1 stably finite strongly connected strongly $\pi$-regular strongly regular symmetric T-nilpotent radical (left) T-nilpotent radical (right) top regular top simple top simple Artinian uniform (left) uniform (right) unit regular valuation ring (left) valuation ring (right) von Neumann regular V ring (left) V ring (right) weakly clean Zorn