motivation for Euclidean domains

motivation for Euclidean domains

UFDs, PIDs, and

Euclidean domains

are ways of building successively more of standard number into a domain.

First, observe that the units are numbers that behave in a manner

similar

to

1

.

Obvious

examples, besides

1

itself, are

-

1

in

ℤ

or

i

and

-

i

in the

Gaussian integers

.

Ideals behave something like the set of products of an element; in

ℤ

those are the ideals (together with the

zero ideal

), and ideals in other rings have some similar

behavior

.

In

commutative rings

,

prime ideals

have one

property

similar to the

prime numbers

. Specifically, a product of two elements is in the ideal exactly when one of those elements is already in the ideal. For example, in

ℤ

, if

a

⋅

b

is even, we know that either

a

or

b

is even, but if

a

⋅

b

is a

multiple

of four, we do not know that

a

or

b

is a multiple of four, since they could both be

even numbers

that are not multiples of four.

The other property most associated with prime numbers is their irreducibility: The only way to factor an

irreducible element

is to use a unit, and since units are “

1

-like”, that does not really break the element into smaller pieces. (Specifically, the non-unit factor can always be broken into another unit times the original irreducible element.)

In a UFD, these two properties of prime numbers coincide for nonzero numbers. All

prime elements

(

generators

of prime ideals) are

irreducible

, and all irreducibles are prime elements. Also, all numbers can be factored into prime elements the same way

integers

can be factored into primes.

A PID behaves even more like the integers by adding the

concept

of a

greatest common divisor

. Formally, this holds because, for any two ideals in any ring, we can find a minimal ideal which contains both of them, and in a PID we have the guarantee that the new ideal is generated by a particular element–the greatest common divisor. The

Dedekind-Hasse valuation

on the ring encodes this property by requiring that, if

a

is not a multiple of

b

(that is, in

⟨

b

⟩

), then there is a common

divisor

which is simpler than

b

. (The formal

definition

is that the element be of the form

a

⁢

x

+

b

⁢

y

, but there is, in general, a between

linear combinations

of elements and their greatest common divisor.)

Being a Euclidean domain is an even stronger requirement, the most important effect of which is to provide Euclid’s algorithm for finding gcd’s. The key property is that

division

with

remainders

can be performed akin to the way it is done in the integers. A

Euclidean valuation

again encodes this property by ensuring that remainders are limited (specifically, requiring that the norm of the remainder be less than the norm of the divisor). This the remainders to get successively smaller, guaranteeing that the process

eventually

terminates.

Title

motivation for Euclidean domains

Canonical

name

MotivationForEuclideanDomains

Date of creation

2013-03-22 12:52:05

Last modified on

2013-03-22 12:52:05

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

8

Author

Wkbj79 (1863)

Entry type

Topic

Classification

msc 13G05

Related topic

EuclideanRing

Related topic

PID

Related topic

UFD

Related topic

IntegralDomain

Generated on Fri Feb 9 17:32:54 2018 by

LaTeXML