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'zero ideal'
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| Title of object: |
zero ideal |
| Canonical Name: |
ZeroIdeal |
| Type: |
Definition |
| Created on: |
2009-01-18 08:22:04 |
| Modified on: |
2009-01-18 08:30:34 |
| Classification: |
msc:13A15, msc:11N80, msc:16D25, msc:14K99 |
Revision comment (for changes between this and next version):
| Changes for correction #14521 ('Prime'). |
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
\theoremstyle{definition}
\newtheorem*{thmplain}{Theorem}
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Content:
The subset $\{0\}$ of a ring $R$ is the least two-sided ideal of $R$.\, As a principal ideal, it is often denoted by
$$(0)$$
and called the {\em zero ideal}.\\
The zero ideal is the identity element in the addition of ideals and the absorbing element in the \PMlinkname{multiplication of ideals}{ProductOfIdeals}.\, The quotient ring $R/(0)$ is trivially isomorphic to $R$.
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