Description: The polynomial ring $\mathbb Z[x]$ over the integers (1)

Notes: All prime ideals are $1$-or-$2$ generated, but $(3,x,y)$ is not. Krull dimension 2.

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
coherent (left)
coherent (right)
commutative
connected
CS (left)
CS (right)
DCC annihilator (left)
DCC annihilator (right)
Dedekind finite
domain
duo (left)
duo (right)
finite uniform dimension (left)
finite uniform dimension (right)
finitely generated socle (left)
finitely generated socle (right)
Goldie (left)
Goldie (right)
IBN
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
lift/rad
NI (nilpotents form an ideal)
nil radical
nilpotent radical
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
quasi-continuous (left)
quasi-continuous (right)
quasi-duo (left)
quasi-duo (right)
reduced
reversible
Rickart (left)
Rickart (right)
semicommutative (SI condition, zero-insertive)
semiprime
semiprimitive
stably finite
strongly connected
symmetric
T-nilpotent radical (left)
T-nilpotent radical (right)
uniform (left)
uniform (right)

The ring lacks the following properties:

$\pi$-regular
$I_0$
Artinian (left)
Artinian (right)
Bezout (left)
Bezout (right)
Bezout domain (left)
Bezout domain (right)
clean
cogenerator ring (left)
cogenerator ring (right)
cohopfian (left)
cohopfian (right)
continuous (left)
continuous (right)
division ring
dual (left)
dual (right)
essential socle (left)
essential socle (right)
exchange
FI-injective (left)
FI-injective (right)
finite
finitely cogenerated (left)
finitely cogenerated (right)
free ideal ring (left)
free ideal ring (right)
Frobenius
fully prime
fully semiprime
hereditary (left)
hereditary (right)
Kasch (left)
Kasch (right)
local
nonzero socle (left)
nonzero socle (right)
perfect (left)
perfect (right)
primary
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-Frobenius
self-injective (left)
self-injective (right)
semi free ideal ring
semihereditary (left)
semihereditary (right)
semilocal
semiperfect
semiprimary
semiregular
semisimple
serial (left)
serial (right)
simple
simple Artinian
simple socle (left)
simple socle (right)
strongly $\pi$-regular
strongly regular
top regular
top simple
top simple Artinian
unit regular
V ring (left)
V ring (right)
valuation ring (left)
valuation ring (right)
von Neumann regular
Zorn

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