0.1 Divisibility in a monoid
In a commutative monoid , one can speak of divisibility: its element is divisible by its element , iff where . An element of , distinct from the unity of , is called a prime element of , when is divisible only by itself and . The monoid has a unique prime factorisation, if every element of can be presented as a finite product 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. Two elements of have always a greatest common factor. 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 . We say that the integral domain has a divisor theory, if there is a commutative monoid with unique prime factorisation and a homomorphism from the monoid into the monoid , such that the following three properties are true:
A divisibility (http://planetmath.org/DivisibilityInRings) in is valid iff the divisibility is valid in .
If the elements and of are divisible by an element of , then also are divisible by (‘‘’’ means that ; in , 0 is divisible by every element of ).
If , then .
By 1, it is easily seen that two principal divisors and are equal iff the elements and are associates of each other. Especially, the units of determine the unit divisor .
0.3 Uniqueness theorems
Theorem 1. An integral domain has at most one divisor theory. In other words, for any pair of divisor theories and , there is an isomorphism such that always when the principal divisors and correspond to the same element of .
Theorem 2. An integral domain is a unique factorisation domain (http://planetmath.org/UFD) if and only if
has a divisor theory in which all divisors are principal divisors.
Theorem 3. If the divisor theory comprises only a finite number of prime divisors, then is a unique factorisation domain.
The proofs of those theorems are found in , which is available also in Russian (original), English and French.
- 1 S. Borewicz & I. Safarevic: Zahlentheorie. Birkhäuser Verlag. Basel und Stuttgart (1966).
- 2 М. М. Постников: Введение в теорию алгебраических чисел. Издательство ‘‘Наука’’. Москва (1982).
|Date of creation||2013-03-22 17:59:03|
|Last modified on||2013-03-22 17:59:03|
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