sum of ideals
Definition. Let’s consider some set of ideals (left, right or twosided) 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 “$\sum $” as usually.
It is not difficult to be persuaded of the following:

•
The sum of a finite amount of ideals is
$${\U0001d51e}_{1}+{\U0001d51e}_{2}+\mathrm{\cdots}+{\U0001d51e}_{k}=\{{a}_{1}+{a}_{2}+\mathrm{\cdots}+{a}_{k}\mathrm{\vdots}\mathit{\hspace{1em}}{a}_{i}\in {\U0001d51e}_{i}\forall i\}.$$ 
•
The sum of any set of ideals consists of all finite sums $\sum _{j}}{a}_{j$ where every ${a}_{j}$ belongs to one ${\U0001d51e}_{j}$ 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 $\U0001d51e+\U0001d51f=\U0001d521$ in a ring $R$. Because $\U0001d51e\subseteq \U0001d521$ and $\U0001d51f\subseteq \U0001d521$, we can say that $\U0001d521$ is a of both $\U0001d51e$ and $\U0001d51f$.^{1}^{1}This may be motivated by the situation in $\mathbb{Z}$: $(n)\subseteq (m)$ iff $m$ is a factor of $n$. Moreover, $\U0001d521$ is contained in every common factor $\U0001d520$ of $\U0001d51e$ and $\U0001d51f$ by virtue of its minimality. Hence, $\U0001d521$ may be called the greatest common divisor^{} of the ideals $\U0001d51e$ and $\U0001d51f$. The notations
$$\U0001d51e+\U0001d51f=\mathrm{gcd}(\U0001d51e,\U0001d51f)=(\U0001d51e,\U0001d51f)$$ 
are used, too.
In an analogous way, the intersection^{} of ideals may be designated as the least common of the ideals.
The by “$\subseteq $” partially ordered set^{} of all ideals of a ring forms a lattice^{}, where the least upper bound of $\U0001d51e$ and $\U0001d51f$ is $\U0001d51e+\U0001d51f$ and the greatest lower bound^{} is $\U0001d51e\cap \U0001d51f$. See also the example 3 in algebraic lattice.
Title  sum of ideals 
Canonical name  SumOfIdeals 
Date of creation  20130322 14:39:26 
Last modified on  20130322 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 multiple^{} of ideals 