Ring detail


Name: $\mathbb R[x]/(x^2)$

Description: The quotient of the real polynomial ring $\mathbb R[x]$ by the ideal generated by $x^2$

Notes: Krull dimension 0

Keywords quotient ring polynomial ring

Reference(s):

  • (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) Bezout (left) Bezout (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) 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) Frobenius Goldie (left) Goldie (right) I_0 IBN Kasch (left) Kasch (right) lift/rad local nil radical nilpotent radical Noetherian (left) Noetherian (right) nonzero socle (left) nonzero socle (right) Ore ring (left) Ore ring (right) orthogonally finite perfect (left) perfect (right) polynomial identity primary 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 reversible self-injective (left) self-injective (right) semilocal semiperfect semiprimary semiregular serial (left) serial (right) simple socle (left) simple socle (right) simple-injective (left) simple-injective (right) stable range 1 stably finite strongly $\pi$-regular symmetric T-nilpotent radical (left) T-nilpotent radical (right) top regular top simple top simple Artinian uniform (left) uniform (right) valuation ring (left) valuation ring (right) Zorn
    We don't know if the ring has or lacks the following properties: