Description: Ring of formal power series over a field $k$

Notes:

Keywords power series ring

Reference(s):

This ring has the following properties:

$I_0$
2-primal
Abelian
ACC annihilator (left)
ACC annihilator (right)
ACC principal (left)
ACC principal (right)
Baer
Bezout (left)
Bezout (right)
Bezout domain (left)
Bezout domain (right)
clean
coherent (left)
coherent (right)
commutative
connected
CS (left)
CS (right)
DCC annihilator (left)
DCC annihilator (right)
Dedekind finite
distributive (left)
distributive (right)
domain
duo (left)
duo (right)
exchange
finite uniform dimension (left)
finite uniform dimension (right)
finitely generated socle (left)
finitely generated socle (right)
free ideal ring (left)
free ideal ring (right)
Goldie (left)
Goldie (right)
hereditary (left)
hereditary (right)
IBN
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
lift/rad
local
NI (nilpotents form an ideal)
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
principal ideal domain (left)
principal ideal domain (right)
principal ideal ring (left)
principal ideal ring (right)
quasi-continuous (left)
quasi-continuous (right)
quasi-duo (left)
quasi-duo (right)
reduced
reversible
Rickart (left)
Rickart (right)
semi free ideal ring
semicommutative (SI condition, zero-insertive)
semihereditary (left)
semihereditary (right)
semilocal
semiperfect
semiprime
semiregular
serial (left)
serial (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
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)
Frobenius
fully prime
fully semiprime
Kasch (left)
Kasch (right)
nil radical
nilpotent radical
nonzero socle (left)
nonzero socle (right)
perfect (left)
perfect (right)
primary
primitive (left)
primitive (right)
principally injective (left)
principally injective (right)
pseudo-Frobenius (left)
pseudo-Frobenius (right)
quasi-Frobenius
self-injective (left)
self-injective (right)
semiprimary
semiprimitive
semisimple
simple
simple Artinian
simple socle (left)
simple socle (right)
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: