Finally, R is a field if and only if is a principal ideal domain.

Scholarship on color vision has proceeded in three principal domains within the last twenty years.

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 local principal ideal domain, and not a field.

Similar statements hold for any principal ideal domain.

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

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

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

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

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