Theorems · Theorem · commutative algebra
irreducible_iff
∀ {M : Type u_1} [inst : Monoid M] {p : M}, Irreducible p ↔ ¬IsUnit p ∧ ∀ ⦃a b : M⦄, p = a * b → IsUnit a ∨ IsUnit b- Defined in
- Mathlib.Algebra.Group.Irreducible.Defs
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 7 from the axioms · uses no axioms
- Assumes
- Monoid
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Monoidstatement and proof · cited by 3,887
- IsUnitstatement and proof · cited by 1,602
- Irreduciblestatement and proof · cited by 496
- Irreducible.not_isUnitproof · cited by 42
- Irreducible.isUnit_or_isUnitproof · cited by 28
Cited by5
Results whose statement or proof uses this declaration.
- DvdNotUnit.isUnit_of_irreducible_rightproof · cited by 2
- Polynomial.Monic.irreducible_iff_irreducible_map_fraction_mapproof · cited by 2
- Ideal.irreducible_of_irreducible_absNormproof · cited by 1
- RatFunc.irreducible_min_polynomial_valuation_lt_one_and_ne_zeroproof · cited by 1
- Cardinal.not_irreducible_of_aleph0_leproof · cited by 0