sum of ideals

Definition.  Let’s consider some set of ideals (left, right or two-sided) of a ring.  The sum of the ideals is the smallest ideal of the ring containing all those ideals.  The sum of ideals is denoted by using “+” and “” as usually.

It is not difficult to be persuaded of the following:

  • The sum of a finite amount of ideals is

  • The sum of any set of ideals consists of all finite sums jaj where every aj belongs to one 𝔞j of those ideals.

Thus, one can say that the sum ideal is generated by the set of all elements of the individual ideals; in fact it suffices to use all generators of these ideals.

Let  𝔞+𝔟=𝔡  in a ring R.  Because  𝔞𝔡  and  𝔟𝔡,  we can say that 𝔡 is a of both 𝔞 and 𝔟.11This may be motivated by the situation in :  (n)(m)  iff  m is a factor of n.  Moreover, 𝔡 is contained in every common factor 𝔠 of 𝔞 and 𝔟 by virtue of its minimality.  Hence, 𝔡 may be called the greatest common divisorMathworldPlanetmathPlanetmath of the ideals 𝔞 and 𝔟.  The notations


are used, too.

In an analogous way, the intersectionMathworldPlanetmath of ideals may be designated as the least common of the ideals.

The by “partially ordered setMathworldPlanetmath of all ideals of a ring forms a latticeMathworldPlanetmathPlanetmath, where the least upper bound of 𝔞 and 𝔟 is  𝔞+𝔟  and the greatest lower boundMathworldPlanetmath is  𝔞𝔟.  See also the example 3 in algebraic lattice.

Title sum of ideals
Canonical name SumOfIdeals
Date of creation 2013-03-22 14:39:26
Last modified on 2013-03-22 14:39:26
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 22
Author pahio (2872)
Entry type Definition
Classification msc 13C99
Classification msc 16D25
Classification msc 08A99
Synonym greatest common divisor of ideals
Related topic QuotientOfIdeals
Related topic ProductOfIdeals
Related topic LeastCommonMultiple
Related topic TwoGeneratorProperty
Related topic Submodule
Related topic AlgebraicLattice
Related topic LatticeOfIdeals
Related topic MaximalIdealIsPrime
Related topic AnyDivisorIsGcdOfTwoPrincipalDivisors
Related topic GcdDomain
Defines sum ideal
Defines sum of the ideals
Defines addition of ideals
Defines factor of ideal
Defines greatest common divisor of ideals
Defines least common multipleMathworldPlanetmath of ideals