Euclidean domain ⇒ PID
Every Euclidean domain has all ideals principal.
Euclidean domain ⇒ PID: If is a Euclidean domain, then is a principal ideal domain: every ideal of is a principal ideal.
A Euclidean domain is an integral domain equipped with a function such that for all and , there exist with and either or .
A principal ideal domain (PID) is an integral domain such that every ideal is principal, i.e. for some .
PIDs provide strong control of divisibility and modules, and they sit between Euclidean domains and unique factorization: every Euclidean domain is a PID, and every PID is a UFD (see PID implies UFD).
An ideal in a ring is an additive subgroup such that it is stable under multiplication by elements of on one side:
- is a left ideal if for all ,
- is a right ideal if for all .
A principal ideal in a commutative ring is an ideal of the form
for some , i.e. an instance of an ideal generated by one element.