Unique factorization domain
In mathematics, a unique factorization domain (UFD) is, roughly speaking, a ring in which every element can be uniquely written as a product of prime elements, analogous to the fundamental theorem of arithmetic for the integers.Formally, a unique factorization domain is defined to be an integral domain R in which every non-zero non-unit[?] x of R can be written as a product of irreducible elements of R:
- x = p1 p2 ... pn
- x = q1 q2 ... qm,
The uniqueness part is sometimes hard to verify, which is why the following equivalent definition is useful: a unique factorization domain is an integral domain R in which every non-zero non-unit can be written as a product of prime elements of R.
Examples
All principal ideal domains are UFD's; this includes the integers, all fields, all polynomial rings K[X] where K is a field, and the Gaussian integers Z[i].
In general, if R is a UFD, then so is the polynomial ring R[X]. By induction, we therefore see that the polynomial rings Z[X1,...,Xn] as well as K[X1,...,Xn] (K a field) are UFD's.
The formal power series ring K[[X1,...,Xn]] over a field K is also a unique factorization domain.
Properties
In UFD's, every irreducible element is prime (the converse is true in any integral domain).
Any two (or finitely many) elements of a UFD have a greatest common divisor and a least common multiple. Here, a greatest common divisor of a and b is an element d which divides both a and b, and such that every other common divisor of a and b divides d. All greatest common divisors of a and b are associated.
D.W.]--as of
them standing; nor endure my own sweat; nor quench my thirst either with
after dinner; and I should be as uneasy without my gloves as without my
could not lie without a canopy and curtains, as if they were essential
after the German fashion, very incommodiously; I foul them more than the
I complain that they did not keep up the fashion, begun after the example
We are told of that laborious soldier Marius that, growing old, he became
I, in like manner, have suffered myself to fancy a certain form of
a strange common hand: all metal offends me in comparison of a clear and
I owe several other such niceties to custom. Nature has also, on the
than two full meals in one day, without overcharging my stomach, nor a
wind, drying up my mouth, and dulling my appetite; the finding great
marches, which often happen to be all night long, after five or six.
On
wordlookup.net
All is still licensed under the GNU FDL.
It uses material from the wikipedia.
|
|