Definition integrity := forall a b : S, Mult a b = O -> {a = O} + {b = O}.
Definition is_unitary_commutative_ring :=
is_ring /\ commutativity S Mult /\ neutral S A Mult I.
(* Division *)
Definition divide (a b : S) :=
A a /\ A b /\ (b = O \/ a <> O /\ (exists q : S, A q /\ b = Mult a q)).

Theorem div_opp : is_ring -> forall a d : S, divide d a -> divide d (Opp a).
Definition is_gcd (a b d : S) :=
divide d a /\
divide d b /\ (forall q : S, divide q a -> divide q b -> divide q d).