ImageIdealOfDivisor 2008-05-06 14:59:29 2008-05-07 12:09:02 Theorem divisor theory image ideal ideal determined by the divisor % 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} \usepackage[T2A]{fontenc} \usepackage[russian, english]{babel} % 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} \textbf{Theorem.}\, If an integral domain $\mathcal{O}$ has a divisor theory \,$\mathcal{O}^* \to \mathfrak{D}$,\, then the subset $[\mathfrak{a}]$ of $\mathcal{O}$, consisting of 0 and all elements divisible by a divisor $\mathfrak{a}$, is an ideal of $\mathcal{O}$.\, The mapping $$\mathfrak{a} \mapsto [\mathfrak{a}]$$ from the set $\mathfrak{D}$ of divisors into the set of ideals of $\mathcal{O}$ is injective and maps any principal divisor $(\alpha)$ to the principal ideal $(\alpha)$.\\ {\em Proof.}\, Let\, $\alpha,\,\beta \in [\mathfrak{a}]$\, and\, $\lambda \in \mathcal{O}$.\, Then, by the postulate 2 of \PMlinkname{divisor theory}{DivisorTheory}, $\alpha\!-\!\beta$ is divisible by $\mathfrak{a}$ or is 0, and in both cases belongs to $[\mathfrak{a}]$.\, When\, $\lambda\alpha \neq 0$,\, we can write\, $(\alpha) = \mathfrak{ac}$\, with $\mathfrak{c}$ a divisor.\, According to the homomorphicity of the mapping \,$\mathcal{O}^* \to \mathfrak{D}$,\, we have $$(\lambda\alpha) = (\lambda)(\alpha) = (\lambda)\mathfrak{ac},$$ and therefore the element $\lambda\alpha$ is divisible by $\mathfrak{a}$, i.e. $\lambda\alpha \in [\mathfrak{a}]$.\, Thus, $[\mathfrak{a}]$ is an ideal of $\mathcal{O}$. The injectivity of the mapping\, $\mathfrak{a} \mapsto [\mathfrak{a}]$\, follows from the postulate 3 of \PMlinkname{divisor theory}{DivisorTheory}.\\ The ideal $[\mathfrak{a}]$ may be called the {\em image ideal} of $\mathfrak{a}$ or the {\em ideal determined by the divisor} $\mathfrak{a}$.\\ \textbf{Remark.}\, There are integral domains $\mathcal{O}$ having a divisor theory but also having ideals which are not of the form $[\mathfrak{a}]$ (for example a polynomial ring in two indeterminates and its ideal formed by the polynomials without constant term).\, Such rings have ``too many ideals''.\; On the other hand, in some integral domains the monoid of principal ideals cannot be embedded into a free monoid; thus those rings cannot have a divisor theory. \begin{thebibliography}{9} \bibitem{MMP} \CYRM. \CYRM. \CYRP\cyro\cyrs\cyrt\cyrn\cyri\cyrk\cyro\cyrv: {\em \CYRV\cyrv\cyre\cyrd\cyre\cyrn\cyri\cyre\, \cyrv\, \cyrt\cyre\cyro\cyrr\cyri\cyryu\, \cyra\cyrl\cyrg\cyre\cyrb\cyrr\cyra\cyri\cyrch\cyre\cyrs\cyrk\cyri\cyrh \, \cyrch\cyri\cyrs\cyre\cyrl}. \,\CYRI\cyrz\cyrd\cyra\cyrt\cyre\cyrl\cyrsftsn\cyrs\cyrt\cyrv\cyro \, ``\CYRN\cyra\cyru\cyrk\cyra''. \CYRM\cyro\cyrs\cyrk\cyrv\cyra \,(1982). \end{thebibliography}