|
|
|
Viewing Version
16
of
'sum of ideals'
|
[ view 'sum of ideals'
|
back to history
]
| Title of object: |
sum of ideals |
| Canonical Name: |
SumOfIdeals |
| Type: |
Definition |
| Created on: |
2004-09-29 17:18:30 |
| Modified on: |
2008-04-16 04:43:46 |
| Classification: |
msc:08A99, msc:16D25, msc:13C99 |
| Defines: |
sum ideal, addition of ideals, factor of ideal, greatest common divisor of ideals, least common multiple of ideals |
| Synonyms: |
sum of ideals=greatest common divisor of ideals |
Revision comment (for changes between this and next version):
| Changes for correction #14214 ('a couple of suggestions'). |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here |
Content:
\textbf{Definition.}\, Let's consider some set of ideals (left, right or two-sided) of a ring.\, The {\em sum of 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}
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 {\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.
|
|
|
|
|
|
