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).
Let be a ring and an additive subgroup. Then is a left ideal if for every , a right ideal if for every , and a two-sided ideal if both conditions hold.
A principal ideal in a commutative unital ring is an ideal of the form
for some , i.e. an instance of an ideal generated by one element.