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Homesum of ideals
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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 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 factor or divisor of both $\mathfrak{a}$ and $\mathfrak{b}$.^{1}^{1}This 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 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.
Mathematics Subject Classification
13C99 no label found16D25 no label found08A99 no label found Forums
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