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Principal ideal rings and principal ideal domains #

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A principal ideal ring (PIR) is a ring in which all left ideals are principal. A principal ideal domain (PID) is an integral domain which is a principal ideal ring.

Main definitions #

Note that for principal ideal domains, one should use [is_domain R] [is_principal_ideal_ring R]. There is no explicit definition of a PID. Theorems about PID's are in the principal_ideal_ring namespace.

Main results #

structure submodule.is_principal {R : Type u} {M : Type v} [ring R] [add_comm_group M] [module R M] (S : submodule R M) :

An R-submodule of M is principal if it is generated by one element.

Instances of this typeclass
theorem submodule.is_principal_iff {R : Type u} {M : Type v} [ring R] [add_comm_group M] [module R M] (S : submodule R M) :
@[protected, instance]
def bot_is_principal {R : Type u} {M : Type v} [ring R] [add_comm_group M] [module R M] :
@[protected, instance]
structure is_principal_ideal_ring (R : Type u) [ring R] :

A ring is a principal ideal ring if all (left) ideals are principal.

Instances of this typeclass
noncomputable def submodule.is_principal.generator {R : Type u} {M : Type v} [add_comm_group M] [ring R] [module R M] (S : submodule R M) [S.is_principal] :

generator I, if I is a principal submodule, is an x ∈ M such that span R {x} = I

theorem is_prime.to_maximal_ideal {R : Type u} [comm_ring R] [is_domain R] [is_principal_ideal_ring R] {S : ideal R} [hpi : S.is_prime] (hS : S ) :
theorem mod_mem_iff {R : Type u} [euclidean_domain R] {S : ideal R} {x y : R} (hy : y S) :
x % y S x S
noncomputable def principal_ideal_ring.factors {R : Type u} [comm_ring R] [is_domain R] [is_principal_ideal_ring R] (a : R) :

factors a is a multiset of irreducible elements whose product is a, up to units

theorem principal_ideal_ring.mem_submonoid_of_factors_subset_of_units_subset {R : Type u} [comm_ring R] [is_domain R] [is_principal_ideal_ring R] (s : submonoid R) {a : R} (ha : a 0) (hfac : (b : R), b principal_ideal_ring.factors a b s) (hunit : (c : Rˣ), c s) :
a s
theorem principal_ideal_ring.ring_hom_mem_submonoid_of_factors_subset_of_units_subset {R : Type u_1} {S : Type u_2} [comm_ring R] [is_domain R] [is_principal_ideal_ring R] [semiring S] (f : R →+* S) (s : submonoid S) (a : R) (ha : a 0) (h : (b : R), b principal_ideal_ring.factors a f b s) (hf : (c : Rˣ), f c s) :
f a s

If a ring_hom maps all units and all factors of an element a into a submonoid s, then it also maps a into that submonoid.

@[protected, instance]

A principal ideal domain has unique factorization

theorem submodule.is_principal.of_comap {R : Type u} {M : Type v} {N : Type u_2} [ring R] [add_comm_group M] [add_comm_group N] [module R M] [module R N] (f : M →ₗ[R] N) (hf : function.surjective f) (S : submodule R N) [hI : (submodule.comap f S).is_principal] :
theorem ideal.is_principal.of_comap {R : Type u} {S : Type u_1} [ring R] [ring S] (f : R →+* S) (hf : function.surjective f) (I : ideal S) [hI : submodule.is_principal (ideal.comap f I)] :

The surjective image of a principal ideal ring is again a principal ideal ring.

theorem span_gcd {R : Type u} [comm_ring R] [is_domain R] [is_principal_ideal_ring R] [gcd_monoid R] (x y : R) :
theorem gcd_dvd_iff_exists {R : Type u} [comm_ring R] [is_domain R] [is_principal_ideal_ring R] [gcd_monoid R] (a b : R) {z : R} :
gcd_monoid.gcd a b z (x y : R), z = a * x + b * y
theorem exists_gcd_eq_mul_add_mul {R : Type u} [comm_ring R] [is_domain R] [is_principal_ideal_ring R] [gcd_monoid R] (a b : R) :
(x y : R), gcd_monoid.gcd a b = a * x + b * y

Bézout's lemma

theorem is_coprime_of_dvd {R : Type u} [comm_ring R] [is_domain R] [is_principal_ideal_ring R] [gcd_monoid R] (x y : R) (nonzero : ¬(x = 0 y = 0)) (H : (z : R), z nonunits R z 0 z x ¬z y) :
theorem dvd_or_coprime {R : Type u} [comm_ring R] [is_domain R] [is_principal_ideal_ring R] [gcd_monoid R] (x y : R) (h : irreducible x) :
theorem is_coprime_of_irreducible_dvd {R : Type u} [comm_ring R] [is_domain R] [is_principal_ideal_ring R] [gcd_monoid R] {x y : R} (nonzero : ¬(x = 0 y = 0)) (H : (z : R), irreducible z z x ¬z y) :
theorem is_coprime_of_prime_dvd {R : Type u} [comm_ring R] [is_domain R] [is_principal_ideal_ring R] [gcd_monoid R] {x y : R} (nonzero : ¬(x = 0 y = 0)) (H : (z : R), prime z z x ¬z y) :
theorem prime.coprime_iff_not_dvd {R : Type u} [comm_ring R] [is_domain R] [is_principal_ideal_ring R] [gcd_monoid R] {p n : R} (pp : prime p) :
theorem irreducible.coprime_pow_of_not_dvd {R : Type u} [comm_ring R] [is_domain R] [is_principal_ideal_ring R] [gcd_monoid R] {p a : R} (m : ) (hp : irreducible p) (h : ¬p a) :
is_coprime a (p ^ m)
theorem irreducible.coprime_or_dvd {R : Type u} [comm_ring R] [is_domain R] [is_principal_ideal_ring R] [gcd_monoid R] {p : R} (hp : irreducible p) (i : R) :
theorem exists_associated_pow_of_mul_eq_pow' {R : Type u} [comm_ring R] [is_domain R] [is_principal_ideal_ring R] [gcd_monoid R] {a b c : R} (hab : is_coprime a b) {k : } (h : a * b = c ^ k) :
(d : R), associated (d ^ k) a
def non_principals (R : Type u) [comm_ring R] :

non_principals R is the set of all ideals of R that are not principal ideals.

theorem non_principals_zorn {R : Type u} [comm_ring R] (c : set (ideal R)) (hs : c non_principals R) (hchain : is_chain has_le.le c) {K : ideal R} (hKmem : K c) :
(I : ideal R) (H : I non_principals R), (J : ideal R), J c J I

Any chain in the set of non-principal ideals has an upper bound which is non-principal. (Namely, the union of the chain is such an upper bound.)

If all prime ideals in a commutative ring are principal, so are all other ideals.