Ring $R_{ 94 }$

Perfect ring that isn't semiprimary

Description:

Let $S=\mathbb Q[x_1, x_2, x_3,\ldots ]$ be the polynomial ring in countably many variables. Let $I$ be the ideal generated by $\{x_i^2\mid i\in \mathbb N\}\cup\{x_ix_j\mid i, j\in \mathbb N, j\geq 2i\}$. The ring is $R=S/I$.

Keywords polynomial ring quotient ring

Reference(s):

  • (Citation needed)


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