principal ideal domain

R is a local principal ideal domain, and not a field.

More generally, any principal ideal domain that is not a field has dimension 1.

In this case it is in fact a principal ideal domain.

In particular, a commutative principal ideal domain which is not a field has global dimension one.

Similar statements hold for any principal ideal domain.

These are completely classified by the structure theorem, taking Z as the principal ideal domain.

More generally, all nonzero prime ideals are maximal in a principal ideal domain.

R is a principal ideal domain with a unique non-zero prime ideal.

All Euclidean domains are principal ideal domains, but the converse is not true.

Any principal ideal domain (in particular, any field).

Also every principal ideal domain is a unique-factorization domain.

All Euclidean domains and all fields are principal ideal domains.

All principal ideal domains are integrally closed.

The next simplest case is the case when the coefficient ring is a principal ideal domain.

They both noticed it was precisely the extra piece of structure needed to turn an integral domain into a principal ideal domain.

R is a principal ideal domain with a unique irreducible element (up to multiplication by units).

Every principal ideal domain is Noetherian.

The structure theorem for finitely generated modules over a principal ideal domain usually appears in the following two forms.

It can also be used to prove the well known structure theorem for finitely generated modules over a principal ideal domain.

Every primary ideal of a principal ideal domain is an irreducible ideal.

Following the same pattern, we may also define torsion-free ranks of all module (mathematics) over any principal ideal domain 'R'.

In principal ideal domains a near converse holds: every nonzero prime ideal is maximal.

These include the full subcategories of commutative rings, integral domains, principal ideal domains, and fields.

He was the first to prove the existence of principal ideal domains that are not Euclidean domains, being his first example.

Every semisimple ring R which is not just a product of fields is a noncommutative right and left principal ideal domain.