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The ring is
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
Multiple selections can be made in both columns.
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The ring is not
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