image ideal of divisor

Theorem.  If an integral domainMathworldPlanetmath 𝒪 has a divisor theory𝒪*𝔇,  then the subset [𝔞] of 𝒪, consisting of 0 and all elements divisible by a divisorMathworldPlanetmathPlanetmath 𝔞, is an ideal of 𝒪.  The mapping


from the set 𝔇 of divisors into the set of ideals of 𝒪 is injective and maps any principal divisor (α) to the principal idealMathworldPlanetmath (α).

Proof.  Let  α,β[𝔞]  and  λ𝒪.  Then, by the postulate 2 of divisor theory (, α-β is divisible by 𝔞 or is 0, and in both cases belongs to [𝔞].  When  λα0,  we can write  (α)=𝔞𝔠  with 𝔠 a divisor.  According to the homomorphicity of the mapping  𝒪*𝔇,  we have


and therefore the element λα is divisible by 𝔞, i.e. λα[𝔞].  Thus, [𝔞] is an ideal of 𝒪.

The injectivity of the mapping  𝔞[𝔞]  follows from the postulate 3 of divisor theory (

The ideal [𝔞] may be called the image ideal of 𝔞 or the ideal determined by the divisor 𝔞.

Remark.  There are integral domains 𝒪 having a divisor theory but also having ideals which are not of the form [𝔞] (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
Canonical name ImageIdealOfDivisor
Date of creation 2013-03-22 18:02:40
Last modified on 2013-03-22 18:02:40
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Theorem
Classification msc 13A15
Classification msc 13A05
Classification msc 11A51
Defines image ideal
Defines ideal determined by the divisor