Theorems · Theorem · commutative algebra

Module.finite_iff_isArtinianRing

∀ (R : Type u_1) (A : Type u_2) [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : Algebra R A]
  [Algebra.FiniteType R A] [IsArtinianRing R], Module.Finite R A ↔ IsArtinianRing A

If A is a finite type algebra over an Artinian ring R, then A is finite over R if and only if A is an Artinian ring.

Defined in

Mathlib.RingTheory.Jacobson.Artinian

Cited by

2 results in Mathlib

Foundations

Depth 150 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraAlgebra.FiniteTypeIsArtinianRing