Theorems · Theorem · commutative algebra
isFiniteLength_quotient_span_singleton
∀ (R : Type u_1) [inst : CommRing R] [IsNoetherianRing R] [Ring.KrullDimLE 1 R] {x : R},
x ∈ nonZeroDivisors R → IsFiniteLength R (R ⧸ Ideal.span {x})The quotient of a Noetherian ring of krull dimension less than or equal to 1 by a principal ideal
is of finite length.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 102 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites26
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- CommRingstatement and proof · cited by 17,173
- ENatproof · cited by 4,985
- Idealstatement · cited by 4,748
- LE.le.transproof · cited by 3,151
- Submonoidstatement · cited by 3,086
- zero_addproof · cited by 2,366
- HasQuotient.Quotientstatement and proof · cited by 2,301
- Nat.cast_zeroproof · cited by 1,870
- WithBotproof · cited by 1,498
- Ideal.spanstatement and proof · cited by 948
- nonZeroDivisorsstatement and proof · cited by 895
Cited by1
Results whose statement or proof uses this declaration.
- Ring.ord_ne_topproof · cited by 3