sum of idealsSumOfIdeals2004-09-29 17:18:302008-08-30 18:13:47Definitionidealsum idealsum of the idealsaddition of idealsfactor of idealgreatest common divisor of idealsleast common multiple of ideals% this is the default PlanetMath preamble. as your knowledge
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% define commands here\textbf{Definition.}\, Let's consider some set of ideals (left, right or two-sided) of a ring.\, The {\em 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:
\begin{itemize}
\item The sum of a finite amount of ideals is
$$\mathfrak{a}_1+\mathfrak{a}_2+\cdots+\mathfrak{a}_k =
\{a_1\!+\!a_2\!+\!\cdots\!+\!a_k\,\vdots \quad a_i \in \mathfrak{a}_i
\,\,\forall i\}.$$
\item 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.
\end{itemize}
Thus, one can say that the sum ideal is {\em 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 \PMlinkescapetext{{\em factor} or {\em divisor}} of both $\mathfrak{a}$ and $\mathfrak{b}$.\footnote{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 {\em 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 {\em least common \PMlinkescapetext{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.