Description: The subring of $\mathbb Q[x]$ generated by the ideal $(x)$ and the subring $\mathbb Z$.

Notes:

Keywords polynomial ring subring

Reference(s):

This ring has the following properties:

2-primal
Abelian
ACC annihilator (left)
ACC annihilator (right)
Baer
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)
NI (nilpotents form an ideal)
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
stably finite
strongly connected
symmetric
uniform (left)
uniform (right)

The ring lacks the following properties:

$\pi$-regular
ACC principal (left)
ACC principal (right)
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)
free ideal ring (left)
free ideal ring (right)
Frobenius
fully prime
fully semiprime
hereditary (left)
hereditary (right)
Kasch (left)
Kasch (right)
Noetherian (left)
Noetherian (right)
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)
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:

$I_0$
Bezout (left)
Bezout (right)
Bezout domain (left)
Bezout domain (right)
clean
coherent (left)
coherent (right)
continuous (left)
continuous (right)
distributive (left)
distributive (right)
exchange
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
lift/rad
local
nil radical
nilpotent radical
semi free ideal ring
semihereditary (left)
semihereditary (right)
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)
weakly clean
Zorn