Theorems · Theorem · ring theory

isSimpleModule_iff_eq_zero_or_injective

∀ (R : Type u) (M : Type v) [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M],
  IsSimpleModule R M ↔
    Nontrivial M ∧
      ∀ (N : Type v) [inst_3 : AddCommGroup N] [inst_4 : Module R N] (f : M →ₗ[R] N), f = 0 ∨ Function.Injective ⇑f

Defined in: Mathlib.RingTheory.SimpleRing.DivisionRing
Cited by: 0 results in Mathlib
Foundations: Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Ring AddCommGroup Module

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DFunLike.coestatement and proof · cited by 62,936
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Bot.botproof · cited by 4,720
Nontrivialstatement and proof · cited by 2,416
HasQuotient.Quotientproof · cited by 2,301
LinearMap.kerproof · cited by 848

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