Theorems · Theorem · commutative algebra

Module.supportDim_quotSMulTop_succ_le_of_notMem_minimalPrimes

∀ {R : Type u_1} [inst : CommRing R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : Module R M] [Module.Finite R M]
  {x : R},
  (∀ p ∈ (Module.annihilator R M).minimalPrimes, x ∉ p) →
    Module.supportDim R (QuotSMulTop x M) + 1 ≤ Module.supportDim R M

If M is a finite module over a commutative ring R, x ∈ M is not in any minimal prime of M, then dim M/xM + 1 ≤ dim M.

Defined in

Mathlib.RingTheory.KrullDimension.Regular

Cited by

1 results in Mathlib

Foundations

Depth 94 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleModule.Finite