Ring detail

Name: Rational numbers: $\mathbb Q$

Description: Field of fractions of the integers $\mathbb Z$

Notes: smallest ordered field

Keywords ring of quotients equivalence relation


  • (Citation needed)

  • This ring has the following properties:
    $\pi$-regular 2-primal Abelian ACC annihilator (left) ACC annihilator (right) ACC principal (left) ACC principal (right) Artinian (left) Artinian (right) Baer Bezout (left) Bezout (right) Bezout domain (left) Bezout domain (right) clean cogenerator ring (left) cogenerator ring (right) coherent (left) coherent (right) cohopfian (left) cohopfian (right) commutative connected continuous (left) continuous (right) DCC annihilator (left) DCC annihilator (right) Dedekind finite distributive (left) distributive (right) division ring domain dual (left) dual (right) duo (left) duo (right) essential socle (left) essential socle (right) exchange FI-injective (left) FI-injective (right) finite uniform dimension (left) finite uniform dimension (right) finitely cogenerated (left) finitely cogenerated (right) finitely generated socle (left) finitely generated socle (right) finitely pseudo-Frobenius (left) finitely pseudo-Frobenius (right) free ideal ring (left) free ideal ring (right) Frobenius fully prime fully semiprime Goldie (left) Goldie (right) hereditary (left) hereditary (right) I_0 IBN Kasch (left) Kasch (right) lift/rad local nil radical nilpotent radical Noetherian (left) Noetherian (right) nonsingular (left) nonsingular (right) nonzero socle (left) nonzero socle (right) Ore domain (left) Ore domain (right) Ore ring (left) Ore ring (right) orthogonally finite perfect (left) perfect (right) polynomial identity primary prime primitive (left) primitive (right) principal ideal domain (left) principal ideal domain (right) principal ideal ring (left) principal ideal ring (right) principally injective (left) principally injective (right) pseudo-Frobenius (left) pseudo-Frobenius (right) quasi-duo (left) quasi-duo (right) quasi-Frobenius reduced reversible Rickart (left) Rickart (right) self-injective (left) self-injective (right) semi free ideal ring semihereditary (left) semihereditary (right) semilocal semiperfect semiprimary semiprime semiprimitive semiregular semisimple serial (left) serial (right) simple simple Artinian simple socle (left) simple socle (right) simple-injective (left) simple-injective (right) stable range 1 stably finite strongly $\pi$-regular strongly regular symmetric T-nilpotent radical (left) T-nilpotent radical (right) top regular top simple top simple Artinian uniform (left) uniform (right) unit regular V ring (left) V ring (right) valuation ring (left) valuation ring (right) von Neumann regular Zorn
    The ring lacks the following properties:
    We don't know if the ring has or lacks the following properties: