Ring detail


Name: $M_n(F)$: the matrix ring over a field

Description: The ring of $n \times n$ matrices with entries from a field $F$, $n$ a natural number greater than $1$

Notes:

Keywords matrix ring

Reference(s):

  • (Citation needed)


  • This ring has the following properties:
    $\pi$-regular $I_0$ ACC annihilator (left) ACC annihilator (right) ACC principal (left) ACC principal (right) Artinian (left) Artinian (right) Baer Bezout (left) Bezout (right) clean cogenerator ring (left) cogenerator ring (right) coherent (left) coherent (right) cohopfian (left) cohopfian (right) connected continuous (left) continuous (right) CS (left) CS (right) DCC annihilator (left) DCC annihilator (right) Dedekind finite dual (left) dual (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 fully prime fully semiprime Goldie (left) Goldie (right) hereditary (left) hereditary (right) IBN Ikeda-Nakayama (left) Ikeda-Nakayama (right) Kasch (left) Kasch (right) lift/rad nil radical nilpotent radical Noetherian (left) Noetherian (right) nonsingular (left) nonsingular (right) nonzero socle (left) nonzero socle (right) Ore ring (left) Ore ring (right) orthogonally finite perfect (left) perfect (right) polynomial identity primary prime primitive (left) primitive (right) 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-Frobenius Rickart (left) Rickart (right) self-injective (left) self-injective (right) semihereditary (left) semihereditary (right) semilocal semiperfect semiprimary semiprime semiprimitive semiregular semisimple serial (left) serial (right) simple simple Artinian simple-injective (left) simple-injective (right) stable range 1 stably finite strongly $\pi$-regular T-nilpotent radical (left) T-nilpotent radical (right) top regular top simple top simple Artinian unit regular V ring (left) V ring (right) von Neumann regular weakly clean Zorn
    We don't know if the ring has or lacks the following properties: