Ring $R_{ 155 }$

$C^\infty_0(\mathbb R)$: the ring of germs of smooth functions on $\mathbb R$ at $0$

Description:

The ring of germs of smooth functions ($C^\infty$ functions) $\mathbb R\to \mathbb R$ at the point $0$.

Notes: Maximal ideal is idempotent

Keywords germs of functions ring of functions

Reference(s):

  • (Citation needed)


  • Known Properties
    Name
    ACC annihilator
    almost maximal ring
    almost maximal valuation ring
    analytically normal
    analytically unramified
    arithmetical
    Bezout
    catenary
    cohopfian
    complete local
    continuous
    countable
    CS
    DCC annihilator
    distributive
    dual
    essential socle
    FGC
    FI-injective
    finite uniform dimension
    finitely generated socle
    finitely pseudo-Frobenius
    Goldie
    Henselian local
    Ikeda-Nakayama
    J-0
    J-1
    J-2
    Kasch
    linearly compact
    maximal ring
    maximal valuation ring
    nonzero socle
    normal
    principally injective
    quasi-continuous
    semi-Noetherian
    serial
    simple socle
    simple-injective
    uniform
    uniserial ring
    universally catenary
    universally Japanese
    valuation ring
    $\pi$-regular
    $h$-local domain
    ?-ring
    ACC principal
    algebraically closed field
    almost Dedekind domain
    almost maximal domain
    Archimedean field
    Artinian
    atomic domain
    Baer
    Bezout domain
    Boolean
    characteristic 0 field
    cogenerator ring
    Cohen-Macaulay
    coherent
    complete discrete valuation ring
    Dedekind domain
    discrete valuation ring
    division ring
    domain
    Euclidean domain
    Euclidean field
    excellent
    field
    finite
    finitely cogenerated
    free ideal ring
    Frobenius
    fully prime
    fully semiprime
    GCD domain
    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
    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-excellent
    quasi-Frobenius
    rad-nil
    regular
    regular local
    Rickart
    Schreier domain
    self-injective
    semi free ideal ring
    semi-Artinian
    semihereditary
    semiprimary
    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
    Armendariz
    clean
    commutative
    compressible
    Dedekind finite
    directly irreducible
    duo
    exchange
    IBN
    IC ring
    involutive
    lift/rad
    local
    McCoy
    NI ring
    nonsingular
    Ore ring
    orthogonally finite
    polynomial identity
    potent
    quasi-duo
    reduced
    reversible
    semicommutative
    semilocal
    semiperfect
    semiprime
    semiregular
    stable range 1
    stably finite
    strongly connected
    symmetric
    top regular
    top simple
    top simple Artinian
    UGP ring
    weakly clean
    Legend
    • = has the property
    • = does not have the property
    • = information not in database

    (Nothing was retrieved.)

    Name Description
    Idempotents $\{0,1\}$
    Left singular ideal $\{0\}$
    Nilpotents $\{0\}$
    Right singular ideal $\{0\}$