Theorems · Theorem · commutative algebra
IsArtinianRing.isUnit_iff_isLeftRegular
∀ {R : Type u_1} [inst : Semiring R] [IsArtinianRing Rᵐᵒᵖ] {x : R}, IsUnit x ↔ IsLeftRegular x- Defined in
- Mathlib.RingTheory.Artinian.Module
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 32 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SemiringIsArtinianRing
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Semiringstatement and proof · cited by 13,802
- IsUnitstatement and proof · cited by 1,602
- MulOppositestatement and proof · cited by 1,135
- MulOpposite.opproof · cited by 520
- IsArtinianRingstatement and proof · cited by 98
- IsLeftRegularstatement · cited by 97
- IsRightRegularproof · cited by 93
- IsArtinianRing.isUnit_iff_isRightRegularproof · cited by 3
- isRightRegular_opproof · cited by 3
- isUnit_opproof · cited by 1
Cited by2
Results whose statement or proof uses this declaration.
- IsArtinianRing.isUnit_iff_isRegular_of_mulOppositeproof · cited by 2
- IsArtinianRing.nonZeroDivisorsLeft_eq_isUnitSubmonoidproof · cited by 0