Ring detail


Name: $\mathbb Z/(p^k)$, $p$ a prime, $k>1$

Description: Quotient ring of the integers (1) by an ideal $(n)$ where $n=p^k$ for some prime number $p$, natural number $k>1$.

Notes:

Keywords quotient ring

Reference(s):

  • (Citation needed)


  • This ring has the following properties:
    $\pi$-regular $I_0$ 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) CS (left) CS (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 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) IBN Ikeda-Nakayama (left) Ikeda-Nakayama (right) Kasch (left) Kasch (right) lift/rad local NI (nilpotents form an ideal) 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-continuous (left) quasi-continuous (right) quasi-duo (left) quasi-duo (right) quasi-Frobenius reversible self-injective (left) self-injective (right) semicommutative (SI condition, zero-insertive) 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 strongly connected 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) weakly clean Zorn
    We don't know if the ring has or lacks the following properties: