Description: ring of rational polynomials over the finite field of $p$ elements

Notes: Is a field that isn't perfect

Keywords rational polynomial ring

Reference(s):

This ring has the following properties:

$\pi$-regular
$I_0$
2-primal
Abelian
ACC annihilator (left)
ACC annihilator (right)
ACC principal (left)
ACC principal (right)
Artinian (left)
Artinian (right)
Baer
Bezout (left)
Bezout (right)
Bezout domain (left)
Bezout domain (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 uniform dimension (left)
finite uniform dimension (right)
finitely cogenerated (left)
finitely cogenerated (right)
finitely generated socle (left)
finitely generated socle (right)
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
free ideal ring (left)
free ideal ring (right)
Frobenius
fully prime
fully semiprime
Goldie (left)
Goldie (right)
hereditary (left)
hereditary (right)
IBN
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
Kasch (left)
Kasch (right)
lift/rad
local
NI (nilpotents form an ideal)
nil radical
nilpotent radical
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)
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)
semi free ideal ring
semicommutative (SI condition, zero-insertive)
semihereditary (left)
semihereditary (right)
semilocal
semiperfect
semiprimary
semiprime
semiprimitive
semiregular
semisimple
serial (left)
serial (right)
simple
simple Artinian
simple socle (left)
simple socle (right)
simple-injective (left)
simple-injective (right)
stable range 1
stably finite
strongly $\pi$-regular
strongly connected
strongly regular
symmetric
T-nilpotent radical (left)
T-nilpotent radical (right)
top regular
top simple
top simple Artinian
uniform (left)
uniform (right)
unit regular
V ring (left)
V ring (right)
valuation ring (left)
valuation ring (right)
von Neumann regular
weakly clean
Zorn

The ring lacks the following properties:

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