divisor theory
0.1 Divisibility in a monoid
In a commutative monoid , one can speak of divisibility: its element is divisible by its element , iff โโ whereโ .โ An element of , distinct from the unity of , is called a prime element
of , when is divisible only by itself and .โ The monoid has a unique prime factorisation, if every element of can be presented as a finite product
โโ of prime elements and this is unique up to the ; then we may say that is a free monoid on the set of its prime elements.
If the monoid has a unique prime factorisation, is divisible only by itself.โ Two elements of have always a greatest common factor.โ If a product is divisible by a prime element , then at least one of and is divisible by .
0.2 Divisor theory of an integral domain
Let be an integral domain and the set of its non-zero elements; this set forms a commutative monoid (with identity
1) with respect to the multiplication of .โ We say that the integral domain has a divisor theory
, if there is a commutative monoid with unique prime factorisation and a homomorphism
โ โ from the monoid into the monoid , such that the following three properties are true:
-
1.
A divisibility (http://planetmath.org/DivisibilityInRings) in is valid iff the divisibility is valid in .
-
2.
If the elements and of are divisible by an element of , then also are divisible by โ (โโโโโ means thatโ ;โ in , 0 is divisible by every element of ).
-
3.
Ifโ ,โ thenโ .
A divisor theory of is denoted byโ .โ The elements of are called divisors and especially the divisors of the form , whereโ , principal divisors.โ The prime elements of are prime divisors
.
By 1, it is easily seen that two principal divisors and are equal iff the elements and are associates of each other.โ Especially, the units of determine the unit divisor .
0.3 Uniqueness theorems
Theorem 1.โ An integral domain has at most one divisor theory.โ In other words, for any pair of divisor theoriesโ โ andโ , there is an isomorphism โโ such thatโ โ always when the principal divisorsโ โ andโ โ correspond to the same element of .
Theorem 2.โ An integral domain is a unique factorisation domain (http://planetmath.org/UFD) if and only if
has a divisor theoryโ โ in which all divisors are principal divisors.
Theorem 3.โ If the divisor theoryโ โ comprises only a finite number of prime divisors, then is a unique factorisation domain.
The proofs of those theorems are found in [1], which is available also in Russian (original), English and French.
References
- 1 S. Borewicz & I. Safarevic: Zahlentheorie.โ Birkhรคuser Verlag. Basel und Stuttgart (1966).
- 2 ะ. ะ. ะะพััะฝะธะบะพะฒ: ะะฒะตะดะตะฝะธะตโ ะฒโ ัะตะพัะธัโ ะฐะปะณะตะฑัะฐะธัะตัะบะธั โ ัะธัะตะป. โะะทะดะฐัะตะปัััะฒะพโ โโะะฐัะบะฐโโ. ะะพัะบะฒะฐโ(1982).
Title | divisor theory |
Canonical name | DivisorTheory |
Date of creation | 2013-03-22 17:59:03 |
Last modified on | 2013-03-22 17:59:03 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 17 |
Author | pahio (2872) |
Entry type | Definition |
Classification | msc 11A51 |
Classification | msc 13A05 |
Related topic | UniqueFactorizationAndIdealsInRingOfIntegers |
Related topic | IdealDecompositionInDedekindDomain |
Related topic | EisensteinCriterionInTermsOfDivisorTheory |
Related topic | DivisorsInBaseFieldAndFiniteExtensionField |
Related topic | ExponentOfField |
Related topic | ExponentValuation2 |
Related topic | DedekindDomainsWithFinitelyManyPrimesArePIDs |
Defines | divisor |
Defines | prime divisor |
Defines | principal divisor |
Defines | unit divisor |