Ring $R_{ 43 }$

Kasch not semilocal ring

Description:

Let $M$ be the direct sum of $\mathbb Z/p_i\mathbb Z$ where $p_i$ is the $i$'th prime. The ring $R$ is the trivial extension $T(\mathbb Z, M)$ of $M$ by $\mathbb Z$.

Keywords trivial extension

Reference(s):

  • T.-Y. Lam. Exercises in modules and rings. (2007) @ Ex 8.18, p 214


Known Properties
Name
$I_0$
?-ring
ACC annihilator
ACC principal
almost maximal ring
arithmetical
Armendariz
Bezout
catenary
clean
coherent
cohopfian
DCC annihilator
directly irreducible
distributive
exchange
finitely pseudo-Frobenius
J-0
J-1
J-2
Jacobson
max ring
orthogonally finite
potent
principally injective
rad-nil
semi-Noetherian
stable range 1
strongly connected
UGP ring
universally catenary
universally Japanese
weakly clean
Zorn
$\pi$-regular
$h$-local domain
algebraically closed field
almost Dedekind domain
almost maximal domain
almost maximal valuation ring
analytically normal
analytically unramified
Archimedean field
Artinian
atomic domain
Baer
Bezout domain
Boolean
characteristic 0 field
cogenerator ring
Cohen-Macaulay
complete discrete valuation ring
complete local
continuous
CS
Dedekind domain
discrete valuation ring
division ring
domain
dual
Euclidean domain
Euclidean field
excellent
FGC
FI-injective
field
finite
finite uniform dimension
finitely cogenerated
finitely generated socle
free ideal ring
Frobenius
fully prime
fully semiprime
GCD domain
Goldie
Goldman domain
Gorenstein
Grothendieck
Henselian local
hereditary
Ikeda-Nakayama
Krull domain
linearly compact
local
local complete intersection
maximal ring
maximal valuation ring
Mori domain
N-1
N-2
Nagata
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
pseudo-Frobenius
Pythagorean field
quadratically closed field
quasi-continuous
quasi-excellent
quasi-Frobenius
reduced
regular
regular local
Rickart
Schreier domain
self-injective
semi free ideal ring
semi-Artinian
semihereditary
semilocal
semiperfect
semiprimary
semiprime
semiprimitive
semiregular
semisimple
serial
simple
simple Artinian
simple socle
simple-injective
strongly $\pi$-regular
strongly regular
top regular
top simple
top simple Artinian
torch
uniform
unique factorization domain
uniserial domain
uniserial ring
unit regular
V ring
valuation domain
valuation ring
von Neumann regular
2-primal
Abelian
anti-automorphic
commutative
compressible
countable
Dedekind finite
duo
essential socle
IBN
IC ring
involutive
Kasch
lift/rad
McCoy
NI ring
nil radical
nilpotent radical
nonzero socle
Ore ring
polynomial identity
quasi-duo
reversible
semicommutative
stably finite
symmetric
T-nilpotent radical
Legend
  • = has the property
  • = does not have the property
  • = information not in database
Name Measure
cardinality $\aleph_0$
composition length left: $\infty$right: $\infty$
Krull dimension (classical) 1

(Nothing was retrieved.)