Theorems · Theorem · commutative algebra
Ring.ord_of_irreducible
∀ {R : Type u_1} [inst : CommRing R] [IsPrincipalIdealRing R] {ϖ : R}, Irreducible ϖ → Ring.ord R ϖ = 1In a principal ideal ring, the order of vanishing of an irreducible element is 1.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 93 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRingIsPrincipalIdealRing
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- Fieldproof · cited by 7,404
- ENatstatement and proof · cited by 4,985
- HasQuotient.Quotientproof · cited by 2,301
- Ideal.spanproof · cited by 948
- Irreduciblestatement and proof · cited by 496
- Ideal.IsMaximalproof · cited by 452
- Ideal.Quotient.mk_surjectiveproof · cited by 134
- IsPrincipalIdealRingstatement and proof · cited by 131
- Ring.ordstatement · cited by 28
- Ideal.Quotient.fieldproof · cited by 25
- PrincipalIdealRing.isMaximal_of_irreducibleproof · cited by 4
Cited by1
Results whose statement or proof uses this declaration.
- Ring.ord_eq_addValproof · cited by 3