Theorems · Theorem · commutative algebra
irreducible_iff_prime
∀ {M : Type u_1} [inst : CommMonoidWithZero M] [IsCancelMulZero M] [DecompositionMonoid M] {a : M},
Irreducible a ↔ Prime a- Defined in
- Mathlib.Algebra.Prime.Defs
- Cited by
- 15 results in Mathlib
- Foundations
- Depth 15 from the axioms · uses propext
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommMonoidWithZerostatement and proof · cited by 913
- Irreduciblestatement · cited by 496
- Primestatement · cited by 277
- IsCancelMulZerostatement and proof · cited by 177
- Prime.irreducibleproof · cited by 54
- DecompositionMonoidstatement and proof · cited by 39
- Irreducible.primeproof · cited by 4
Cited by15
Results whose statement or proof uses this declaration.
- Ideal.finprod_heightOneSpectrum_factorizationproof · cited by 6
- IsDiscreteValuationRing.exists_primeproof · cited by 3
- IsDiscreteValuationRing.addVal_defproof · cited by 3
- map_prime_of_factor_orderIsoproof · cited by 2
- AddCommGroup.equiv_free_prod_directSum_zmodproof · cited by 2
- DivisorChain.eq_pow_second_of_chain_of_has_chainproof · cited by 2
- IsDedekindDomain.HeightOneSpectrum.mem_integers_of_valuation_le_oneproof · cited by 1
- UniqueFactorizationMonoid.primeFactors_eq_primeFactors_natAbsproof · cited by 1
- GaussianInt.prime_of_nat_prime_of_mod_four_eq_threeproof · cited by 1
- Nat.Prime.sq_add_sqproof · cited by 1
- associates_irreducible_iff_primeproof · cited by 0