End UnitRingTheory.
Arguments invrK {R}.
Arguments invr_inj {R} [x1 x2].
Section UnitRingMorphism.
Variables (R S : unitRingType) (f : {rmorphism R -> S}).

(* Specifications: *)
Lemma rmorph_unit x : x \in unit -> f x \in unit.
Lemma rmorphV : {in unit, {morph f: x / x^-1}}.
Lemma rmorph_div x y : y \in unit -> f (x / y) = f x / f y.
Lemma mulr1_eq x y : x * y = 1 -> x^-1 = y.
Lemma divr1_eq x y : x / y = 1 -> x = y. Proof. by move/mulr1_eq/invr_inj. Qed.
Lemma divKr x : x \is a unit -> {in unit, involutive (fun y => x / y)}.
Lemma scaler_unit k x : k \in unit -> (k *: x \in unit) = (x \in unit).
Lemma invrZ k x : k \in unit -> x \in unit -> (k *: x)^-1 = k^-1 *: x^-1.
Section ClosedPredicates.