Ring detail


Name: Non-Artinian simple ring

Description: Let $k$ be any simple ring of characteristic $0$, and let $d$ be a non-inner derivation on $k$. the differential polynomial ring $R=k[x;d]$.

Notes:

Keywords differential polynomial ring

Reference(s):

  • (Citation needed)


  • We don't know if the ring has or lacks the following properties:
    $\pi$-regular $I_0$ 2-primal Abelian ACC annihilator (left) ACC annihilator (right) ACC principal (left) ACC principal (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) continuous (left) continuous (right) CS (left) CS (right) DCC annihilator (left) DCC annihilator (right) Dedekind finite distributive (left) distributive (right) 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 generated socle (left) finitely generated socle (right) finitely pseudo-Frobenius (left) finitely pseudo-Frobenius (right) free ideal ring (left) free ideal ring (right) Goldie (left) Goldie (right) hereditary (left) hereditary (right) IBN Ikeda-Nakayama (left) Ikeda-Nakayama (right) Kasch (left) Kasch (right) NI (nilpotents form an ideal) 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 polynomial identity principal ideal domain (left) principal ideal domain (right) principal ideal ring (left) principal ideal ring (right) principally injective (left) principally injective (right) quasi-continuous (left) quasi-continuous (right) quasi-duo (left) quasi-duo (right) reduced reversible Rickart (left) Rickart (right) self-injective (left) self-injective (right) semi free ideal ring semicommutative (SI condition, zero-insertive) semihereditary (left) semihereditary (right) semiregular simple socle (left) simple socle (right) simple-injective (left) simple-injective (right) stable range 1 stably finite strongly $\pi$-regular strongly connected strongly regular symmetric top regular top simple uniform (left) uniform (right) unit regular V ring (left) V ring (right) von Neumann regular weakly clean Zorn