Ring $R_{ 99 }$

Custom Krull dimension valuation ring

Description:

By Krull's theorem for valuation rings, we may take $n>1$ and $G=\mathbb Z^n$ with lexicographic order to form a totally ordered Abelian group, and the resulting valuation domain $R$ is the ring. In this construction, is is possible to make choices to ensure the resulting ring is countable.

Notes: The classical Krull dimension of $R$ corresponds to the rank of $G$, in this case $n>1$.

Reference(s):

  • (Citation needed)


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