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Divisor Theory
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 presentation is unique up to the order of the prime factors
; then we may say that
is a free monoid
on the set of its prime elements.
If the monoid
has a unique prime factorisation,
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
.
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
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 addition, 0 is divisible by every element of
).
- If
, then
.
is denoted by
. The elements of
are called divisors and especially the divisors of the form 
, where
, principal divisors. 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. Especially, the units
of
determine the unit divisor
.
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
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 [1], which is available also in Russian (original), English and French.
Bibliography
- 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).