Description: The polynomial ring $F[x]$ over a field $F$

Notes:

Keywords polynomial ring

Reference(s):

This ring has the following properties:

2-primal
Abelian
ACC annihilator (left)
ACC annihilator (right)
ACC principal (left)
ACC principal (right)
Baer
Bezout (left)
Bezout (right)
Bezout domain (left)
Bezout domain (right)
coherent (left)
coherent (right)
commutative
connected
DCC annihilator (left)
DCC annihilator (right)
Dedekind finite
distributive (left)
distributive (right)
domain
duo (left)
duo (right)
finite uniform dimension (left)
finite uniform dimension (right)
finitely generated socle (left)
finitely generated socle (right)
free ideal ring (left)
free ideal ring (right)
Goldie (left)
Goldie (right)
hereditary (left)
hereditary (right)
IBN
Noetherian (left)
Noetherian (right)
nonsingular (left)
nonsingular (right)
Ore domain (left)
Ore domain (right)
Ore ring (left)
Ore ring (right)
orthogonally finite
polynomial identity
prime
principal ideal domain (left)
principal ideal domain (right)
principal ideal ring (left)
principal ideal ring (right)
quasi-duo (left)
quasi-duo (right)
reduced
reversible
Rickart (left)
Rickart (right)
semi free ideal ring
semihereditary (left)
semihereditary (right)
semiprime
stably finite
symmetric
uniform (left)
uniform (right)

The ring lacks the following properties:

$\pi$-regular
Artinian (left)
Artinian (right)
cogenerator ring (left)
cogenerator ring (right)
cohopfian (left)
cohopfian (right)
division ring
dual (left)
dual (right)
essential socle (left)
essential socle (right)
FI-injective (left)
FI-injective (right)
finite
finitely cogenerated (left)
finitely cogenerated (right)
Frobenius
fully prime
fully semiprime
Kasch (left)
Kasch (right)
nonzero socle (left)
nonzero socle (right)
perfect (left)
perfect (right)
primary
primitive (left)
primitive (right)
principally injective (left)
principally injective (right)
pseudo-Frobenius (left)
pseudo-Frobenius (right)
quasi-Frobenius
self-injective (left)
self-injective (right)
semiprimary
semisimple
simple
simple Artinian
simple socle (left)
simple socle (right)
strongly $\pi$-regular
strongly regular
unit regular
V ring (left)
V ring (right)
von Neumann regular

We don't know if the ring has or lacks the following properties:

clean
continuous (left)
continuous (right)
exchange
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
I_0
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
lift/rad
local
nil radical
nilpotent radical
semilocal
semiperfect
semiprimitive
semiregular
serial (left)
serial (right)
simple-injective (left)
simple-injective (right)
stable range 1
T-nilpotent radical (left)
T-nilpotent radical (right)
top regular
top simple
top simple Artinian
valuation ring (left)
valuation ring (right)
Zorn