Theorems · Theorem · commutative algebra
irreducible_sq_not_dvd_iff_eq_zero_and_no_irreducibles_or_squarefree
∀ {R : Type u_1} [inst : CommMonoidWithZero R] [WfDvdMonoid R] (r : R),
(∀ (x : R), Irreducible x → ¬x * x ∣ r) ↔ (r = 0 ∧ ∀ (x : R), ¬Irreducible x) ∨ Squarefree r- Defined in
- Mathlib.Algebra.Squarefree.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 17 from the axioms · uses propext, Classical.choice, Quot.sound
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.
- eq_or_neproof · cited by 1,117
- CommMonoidWithZerostatement and proof · cited by 913
- Irreduciblestatement and proof · cited by 496
- Squarefreestatement and proof · cited by 112
- Irreducible.not_isUnitproof · cited by 42
- WfDvdMonoidstatement and proof · cited by 37
- squarefree_iff_no_irreduciblesproof · cited by 1
Cited by2
Results whose statement or proof uses this declaration.
- squarefree_iff_irreducible_sq_not_dvd_of_exists_irreducibleproof · cited by 1
- squarefree_iff_irreducible_sq_not_dvd_of_ne_zeroproof · cited by 0