image ideal of divisor
from the set of divisors into the set of ideals of is injective and maps any principal divisor to the principal ideal .
Proof. Let and . Then, by the postulate 2 of divisor theory (http://planetmath.org/DivisorTheory), is divisible by or is 0, and in both cases belongs to . When , we can write with a divisor. According to the homomorphicity of the mapping , we have
and therefore the element is divisible by , i.e. . Thus, is an ideal of .
The injectivity of the mapping follows from the postulate 3 of divisor theory (http://planetmath.org/DivisorTheory).
The ideal may be called the image ideal of or the ideal determined by the divisor .
Remark. There are integral domains having a divisor theory but also having ideals which are not of the form (for example a polynomial ring in two indeterminates and its ideal formed by the polynomials without constant term). Such rings have ‘‘too many ideals’’. On the other hand, in some integral domains the monoid of principal ideals cannot be embedded into a free monoid; thus those rings cannot have a divisor theory.
- 1 М. М. Постников: Введение в теорию алгебраических чисел. Издательство ‘‘Наука’’. Москва (1982).
|Title||image ideal of divisor|
|Date of creation||2013-03-22 18:02:40|
|Last modified on||2013-03-22 18:02:40|
|Last modified by||pahio (2872)|
|Defines||ideal determined by the divisor|