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 ``$\sum$ '' as usually.

It is not difficult to be persuaded of the following:

• The sum of a finite amount of ideals is $$\mathfrak{a}_1+\mathfrak{a}_2+\cdots+\mathfrak{a}_k = \{a_1\!+\!a_2\!+\!\cdots\!+\!a_k\,\vdots \quad a_i \in \mathfrak{a}_i \,\,\forall i\}.$$
• The sum of any set of ideals consists of all finite sums $\displaystyle\sum_j a_j$ where every $a_j$ belongs to one $\mathfrak{a}_j$ of those ideals.

Let $\mathfrak{a}+\mathfrak{b} = \mathfrak{d}$ in a ring $R$ . Because $\mathfrak{a} \subseteq \mathfrak{d}$ and $\mathfrak{b} \subseteq \mathfrak{d}$ , we can say that $\mathfrak{d}$ is a factor or divisor of both $\mathfrak{a}$ and $\mathfrak{b}$ .¹ Moreover, $\mathfrak{d}$ is contained in every common factor $\mathfrak{c}$ of $\mathfrak{a}$ and $\mathfrak{b}$ by virtue of its minimality. Hence, $\mathfrak{d}$ may be called the greatest common divisor of the ideals $\mathfrak{a}$ and $\mathfrak{b}$ . The notations $$\mathfrak{a}+\mathfrak{b} = \gcd(\mathfrak{a}, \,\mathfrak{b}) = (\mathfrak{a}, \,\mathfrak{b})$$ are used, too.

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

The by ``$\subseteq$ '' partially ordered set of all ideals of a ring forms a lattice, where the least upper bound of $\mathfrak{a}$ and $\mathfrak{b}$ is $\mathfrak{a+b}$ and the greatest lower bound is $\mathfrak{a\cap b}$ . See also the example 3 in algebraic lattice.

Footnotes

... .¹

This may be motivated by the situation in $\mathbb{Z}$ : $(n) \subseteq (m)$ iff $m$ is a factor of $n$ .