Again, in the language of abstract algebra, the above says that "'Z "'is a Euclidean domain.
22.
The ring H of Hurwitz quaternions is not commutative, hence it is not an actual Euclidean domain, and it does not have principal.
23.
*PM : partial fractions in Euclidean domains, id = 7612-- WP guess : partial fractions in Euclidean domains-- Status:
24.
*PM : partial fractions in Euclidean domains, id = 7612-- WP guess : partial fractions in Euclidean domains-- Status:
25.
Secondly, it is very similar to the case of the integers, and this analogy is the source of the notion of Euclidean domain.
26.
The rings for which such a theorem exists are called Euclidean domains, but in this generality uniqueness of the quotient and remainder are not guaranteed.
27.
The first example of a Euclidean domain that was not norm-Euclidean ( with " D " = 69 ) was published in 1994.
28.
Euclidean domains ( also known as "'Euclidean rings "') are defined as integral domains which support the following generalization of Euclidean division:
29.
This is " false " in general for PIDs, thus providing one of the rare mathematical features of Euclidean domains that do not generalize to all PIDs.
30.
*PM : proof that an Euclidean domain is a PID, id = 3015-- WP guess : proof that an Euclidean domain is a PID-- Status:
