Documentation

Mathlib.RingTheory.Ideal.Nonunits

The set of non-invertible elements of a monoid #

Main definitions #

Main results #

def nonunits (α : Type u_4) [Monoid α] :
Set α

The set of non-invertible elements of a monoid.

Equations
    Instances For
      @[simp]
      theorem mem_nonunits_iff {α : Type u_2} {a : α} [Monoid α] :
      theorem mul_mem_nonunits_right {α : Type u_2} {a b : α} [CommMonoid α] :
      b nonunits αa * b nonunits α
      theorem mul_mem_nonunits_left {α : Type u_2} {a b : α} [CommMonoid α] :
      a nonunits αa * b nonunits α
      theorem zero_mem_nonunits {α : Type u_2} [MonoidWithZero α] :
      0 nonunits α 0 1
      @[simp]
      theorem one_notMem_nonunits {α : Type u_2} [Monoid α] :
      1nonunits α
      @[deprecated one_notMem_nonunits (since := "2025-05-23")]
      theorem one_not_mem_nonunits {α : Type u_2} [Monoid α] :
      1nonunits α

      Alias of one_notMem_nonunits.

      @[simp]
      theorem map_mem_nonunits_iff {F : Type u_1} {α : Type u_2} {β : Type u_3} [Monoid α] [Monoid β] [FunLike F α β] [MonoidHomClass F α β] (f : F) [IsLocalHom f] (a : α) :
      theorem coe_subset_nonunits {α : Type u_2} [Semiring α] {I : Ideal α} (h : I ) :
      I nonunits α
      theorem exists_max_ideal_of_mem_nonunits {α : Type u_2} {a : α} [CommSemiring α] (h : a nonunits α) :
      ∃ (I : Ideal α), I.IsMaximal a I