Theorems · Theorem · ring theory

isSimpleModule_iff_quot_maximal

∀ {R : Type u_2} [inst : Ring R] {M : Type u_4} [inst_1 : AddCommGroup M] [inst_2 : Module R M],
  IsSimpleModule R M ↔ ∃ I, I.IsMaximal ∧ Nonempty (M ≃ₗ[R] R ⧸ I)

A module is simple iff it's isomorphic to the quotient of the ring by a maximal left ideal (not necessarily unique if the ring is not commutative).

Defined in

Mathlib.RingTheory.SimpleModule.Basic

Cited by

5 results in Mathlib

Foundations

Depth 93 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupModule