Ring $R_{ 90 }$

$\mathbb Q[x,y]/(x^2, xy)$

Description:

The quotient ring $\mathbb Q[x,y]/(x^2, xy)$

Notes: The zero ideal is not primary, but it satisfies the false naive definition of primary as being $xy\in I$ implies $x^n\in I$ or $y^n\in I$.

Keywords polynomial ring quotient ring

Reference(s):

  • H. C. Hutchins. Examples of commutative rings. (1981) @ Example 24 p 60
  • I. Kaplansky. Commutative rings. (1974) @ Exercise 6 p 102


Known Properties
Name
almost maximal ring
Armendariz
CS
excellent
finitely pseudo-Frobenius
Grothendieck
J-0
J-1
J-2
Kasch
max ring
quasi-continuous
quasi-excellent
simple-injective
stable range 1
UGP ring
uniform
universally catenary
$\pi$-regular
$h$-local domain
$I_0$
?-ring
algebraically closed field
almost Dedekind domain
almost maximal domain
almost maximal valuation ring
analytically normal
analytically unramified
Archimedean field
arithmetical
Artinian
atomic domain
Baer
Bezout
Bezout domain
Boolean
characteristic 0 field
clean
cogenerator ring
Cohen-Macaulay
cohopfian
complete discrete valuation ring
complete local
continuous
Dedekind domain
discrete valuation ring
distributive
division ring
domain
dual
essential socle
Euclidean domain
Euclidean field
exchange
FGC
FI-injective
field
finite
finitely cogenerated
free ideal ring
Frobenius
fully prime
fully semiprime
GCD domain
Goldman domain
Gorenstein
Henselian local
hereditary
Ikeda-Nakayama
Krull domain
linearly compact
local
local complete intersection
maximal ring
maximal valuation ring
Mori domain
N-1
N-2
nonsingular
normal
normal domain
ordered field
Ore domain
PCI ring
perfect
perfect field
periodic
potent
primary
prime
primitive
principal ideal domain
principal ideal ring
principally injective
Prufer domain
pseudo-Frobenius
Pythagorean field
quadratically closed field
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
strongly $\pi$-regular
strongly regular
top regular
top simple
top simple Artinian
torch
unique factorization domain
uniserial domain
uniserial ring
unit regular
V ring
valuation domain
valuation ring
von Neumann regular
weakly clean
Zorn
2-primal
Abelian
ACC annihilator
ACC principal
anti-automorphic
catenary
coherent
commutative
compressible
countable
DCC annihilator
Dedekind finite
directly irreducible
duo
finite uniform dimension
finitely generated socle
Goldie
IBN
IC ring
involutive
Jacobson
lift/rad
McCoy
Nagata
NI ring
nil radical
nilpotent radical
Noetherian
nonzero socle
Ore ring
orthogonally finite
polynomial identity
quasi-duo
rad-nil
reversible
semi-Noetherian
semicommutative
simple socle
stably finite
strongly connected
symmetric
T-nilpotent radical
universally Japanese
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
Name Description
Idempotents $\{0,1\}$
Left socle $(x)$
Right socle $(x)$