isSimpleRing_isArtinianRing_iff — Mathlib

English

If R is a simple Artinian ring, then R is a semisimple ring.

Русский

Если R простое кольцо с конечной размерностью, то R — полупрямое кольцо.

LaTeX

$$$\\text{IsSimpleRing}(R) \\wedge \\text{IsArtinianRing}(R) \\Rightarrow \\text{IsSemisimpleRing}(R)$$$

Lean4

theorem isSimpleRing_isArtinianRing_iff :
    IsSimpleRing R ∧ IsArtinianRing R ↔ IsSemisimpleRing R ∧ IsIsotypic R R ∧ Nontrivial R :=
  by
  refine ⟨fun ⟨_, _⟩ ↦ ?_, fun ⟨_, _, _⟩ ↦ ?_⟩
  on_goal 1 => have := IsSimpleRing.isSemisimpleRing_iff_isArtinianRing.mpr ‹_›
  all_goals
    simp_rw [isIsotypic_iff_isFullyInvariant_imp_bot_or_top, isFullyInvariant_iff_isTwoSided,
      isSimpleRing_iff_isTwoSided_imp] at *
  · exact ⟨this, by rwa [and_comm]⟩
  · exact ⟨⟨‹_›, ‹_›⟩, inferInstance⟩

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