Ring $R_{ 122 }$

Pseudo-Frobenius, not quasi-Frobenius ring

Description:

Let $B$ be the ring of $2$-adic integers, and $Q$ be its field of quotients. The required ring is the trivial extension $R=T(B, Q/B)$.

Keywords ring of quotients trivial extension

Reference(s):

  • A. V. Mikhalev and G. F. Pilz. The concise handbook of algebra. (2013) @ bottom of p 259


Known Properties
Name
ACC principal
almost maximal ring
almost maximal valuation ring
analytically normal
analytically unramified
arithmetical
Armendariz
Bezout
catenary
coherent
complete local
distributive
FGC
Henselian local
J-0
J-1
J-2
linearly compact
maximal ring
maximal valuation ring
semi-Noetherian
serial
uniserial ring
universally catenary
universally Japanese
valuation ring
$\pi$-regular
$h$-local domain
?-ring
ACC annihilator
algebraically closed field
almost Dedekind domain
almost maximal domain
Archimedean field
Artinian
atomic domain
Baer
Bezout domain
Boolean
characteristic 0 field
Cohen-Macaulay
complete discrete valuation ring
countable
DCC annihilator
Dedekind domain
discrete valuation ring
division ring
domain
Euclidean domain
Euclidean field
excellent
field
finite
free ideal ring
Frobenius
fully prime
fully semiprime
GCD domain
Goldie
Goldman domain
Gorenstein
Grothendieck
hereditary
Jacobson
Krull domain
local complete intersection
max ring
Mori domain
N-1
N-2
Nagata
nil radical
nilpotent radical
Noetherian
nonsingular
normal
normal domain
ordered field
Ore domain
PCI ring
perfect
perfect field
periodic
primary
prime
primitive
principal ideal domain
principal ideal ring
Prufer domain
Pythagorean field
quadratically closed field
quasi-excellent
quasi-Frobenius
rad-nil
reduced
regular
regular local
Rickart
Schreier domain
semi free ideal ring
semi-Artinian
semihereditary
semiprimary
semiprime
semiprimitive
semisimple
simple
simple Artinian
strongly $\pi$-regular
strongly regular
T-nilpotent radical
torch
unique factorization domain
uniserial domain
unit regular
V ring
valuation domain
von Neumann regular
Zorn
$I_0$
2-primal
Abelian
anti-automorphic
clean
cogenerator ring
cohopfian
commutative
compressible
continuous
CS
Dedekind finite
directly irreducible
dual
duo
essential socle
exchange
FI-injective
finite uniform dimension
finitely cogenerated
finitely generated socle
finitely pseudo-Frobenius
IBN
IC ring
Ikeda-Nakayama
involutive
Kasch
lift/rad
local
McCoy
NI ring
nonzero socle
Ore ring
orthogonally finite
polynomial identity
potent
principally injective
pseudo-Frobenius
quasi-continuous
quasi-duo
reversible
self-injective
semicommutative
semilocal
semiperfect
semiregular
simple socle
simple-injective
stable range 1
stably finite
strongly connected
symmetric
top regular
top simple
top simple Artinian
UGP ring
uniform
weakly clean
Legend
  • = has the property
  • = does not have the property
  • = information not in database
Name Measure
Krull dimension (classical) 1
Name Description
Idempotents $\{0,1\}$