Description: Let $D=F[[x]]$ and let $Q$ be the field of fractions of $D$. Set $V=Q/D$. The ring is the trivial extension $(D, V)$

Notes: Krull dimension 2

Keywords trivial extension power series ring

Reference(s):

- Nicholson, W. Keith; Mohamed F. Yousif., Quasi-Frobenius Rings., Vol. 158. Cambridge University Press, (2003). P 133-134 Ex 6.6

This ring has the following properties:

$I_0$
2-primal
Abelian
Bezout (left)
Bezout (right)
clean
cohopfian (left)
cohopfian (right)
commutative
connected
continuous (left)
continuous (right)
CS (left)
CS (right)
Dedekind finite
distributive (left)
distributive (right)
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)
IBN
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
Kasch (left)
Kasch (right)
lift/rad
local
NI (nilpotents form an ideal)
nonzero socle (left)
nonzero socle (right)
Ore ring (left)
Ore ring (right)
orthogonally finite
polynomial identity
principally injective (left)
principally injective (right)
quasi-continuous (left)
quasi-continuous (right)
quasi-duo (left)
quasi-duo (right)
reversible
semicommutative (SI condition, zero-insertive)
semilocal
semiperfect
semiregular
serial (left)
serial (right)
simple socle (left)
simple socle (right)
simple-injective (left)
simple-injective (right)
stable range 1
stably finite
strongly connected
symmetric
top regular
top simple
top simple Artinian
uniform (left)
uniform (right)
valuation ring (left)
valuation ring (right)
weakly clean

The ring lacks the following properties:

$\pi$-regular
ACC annihilator (left)
ACC annihilator (right)
ACC principal (left)
ACC principal (right)
Artinian (left)
Artinian (right)
Baer
Bezout domain (left)
Bezout domain (right)
cogenerator ring (left)
cogenerator ring (right)
DCC annihilator (left)
DCC annihilator (right)
division ring
domain
finite
free ideal ring (left)
free ideal ring (right)
Frobenius
fully prime
fully semiprime
Goldie (left)
Goldie (right)
hereditary (left)
hereditary (right)
nil radical
nilpotent radical
Noetherian (left)
Noetherian (right)
nonsingular (left)
nonsingular (right)
Ore domain (left)
Ore domain (right)
perfect (left)
perfect (right)
primary
prime
primitive (left)
primitive (right)
principal ideal domain (left)
principal ideal domain (right)
principal ideal ring (left)
principal ideal ring (right)
pseudo-Frobenius (left)
pseudo-Frobenius (right)
quasi-Frobenius
reduced
Rickart (left)
Rickart (right)
self-injective (left)
self-injective (right)
semi free ideal ring
semihereditary (left)
semihereditary (right)
semiprimary
semiprime
semiprimitive
semisimple
simple
simple Artinian
strongly $\pi$-regular
strongly regular
T-nilpotent radical (left)
T-nilpotent radical (right)
unit regular
V ring (left)
V ring (right)
von Neumann regular
Zorn

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