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:

The ring lacks the following properties:

ACC principal
algebraically closed field
Artinian
atomic domain
characteristic 0 field
Dedekind domain
Euclidean domain
Euclidean field
field
finite
Frobenius
Noetherian
ordered field
perfect field
principal ideal domain
principal ideal ring
Pythagorean field
quadratically closed field
regular local
semisimple
unique factorization domain
von Neumann regular

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

Bezout domain
Bezout ring
clean
Cohen-Macaulay
coherent
continuous
distributive
dual
finitely pseudo Frobenius
Gorenstein
Jacobson (Hilbert)
Kasch
Krull domain
local
local complete intersection ring
Mori domain
perfect
Prufer domain
pseudo Frobenius
rad-nil
regular
self-injective
semilocal
semiperfect
semiprimary
semiprimitive
semiregular
serial
stable range 1
strongly pi regular
valuation