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Principal ideal rings and principal ideal domains #

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 [integral_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.

def bot_is_principal {R : Type u} {M : Type v} [ring R] [add_comm_group M] [module R M] :
def top_is_principal {R : Type u} [ring R] :
structure is_principal_ideal_ring (R : Type u) [ring R] :

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

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 submodule.is_principal.generator_mem {R : Type u} {M : Type v} [add_comm_group M] [ring R] [module R M] (S : submodule R M) [S.is_principal] :
theorem submodule.is_principal.mem_iff_eq_smul_generator {R : Type u} {M : Type v} [add_comm_group M] [ring R] [module R M] (S : submodule R M) [S.is_principal] {x : M} :
theorem is_prime.to_maximal_ideal {R : Type u} [integral_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

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} [integral_domain R] [is_principal_ideal_ring R] (s : submonoid R) {a : R} (ha : a 0) (hfac : ∀ (b : R), b principal_ideal_ring.factors ab s) (hunit : ∀ (c : units R), 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} [integral_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 af b s) (hf : ∀ (c : units R), 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.


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)] :
theorem is_principal_ideal_ring.of_surjective {R : Type u} {S : Type u_1} [ring R] [ring S] [is_principal_ideal_ring R] (f : R →+* S) (hf : function.surjective f) :

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