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 “ $\\sum$ ” 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.

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 $\\mathfrak{a}+\\mathfrak{b}=\\mathfrak{d}$ in a ring $R$ . Because $\\mathfrak{a}\\subseteq\\mathfrak{d}$ and $\\mathfrak{b}\\subseteq\\mathfrak{d}$ , we can say that $\\mathfrak{d}$ is a of both $\\mathfrak{a}$ and $\\mathfrak{b}$ .¹¹This may be motivated by the situation in $\\mathbb{Z}$ : $(n)\\subseteq(m)$ iff $m$ is a factor of $n$ . 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 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.