Pr\"ufer ringPruferRing2004-01-23 13:30:472008-08-26 17:18:56Theoremfractional ideal of commutative ringPr\"ufer ringcoefficient moduleGaussian ringfractional idealinvertible idealinverse ideal% this is the default PlanetMath preamble. as your knowledge
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\newtheorem{thmplain}{Theorem}\textbf{Definition.}\, A commutative ring $R$ with non-zero unity is a \emph{Pr\"ufer ring} (cf. Pr\"ufer domain) if every finitely generated regular ideal of $R$ is invertible. (It can be proved that if every \PMlinkescapetext{regular} ideal of $R$ generated by two elements is invertible, then all finitely generated \PMlinkescapetext{regular} ideals are invertible; cf. invertibility of regularly generated ideal.)
Denote generally by\, $\mathfrak{m}_p$\, the $R$-module generated by the coefficients of a polynomial $p$ in $T[x]$, where $T$ is the total ring of fractions of $R$.\, Such {\em coefficient modules} are, of course, fractional ideals of $R$.\\
\begin{thmplain}
\,\, (Pahikkala 1982) \, Let $R$ be a commutative ring with non-zero unity and let $T$ be the total ring of fractions of $R$.\, Then, $R$ is a Pr\"ufer ring iff the equation
\begin{align}
\mathfrak{m}_f\mathfrak{m}_g = \mathfrak{m}_{fg}
\end{align}
holds whenever $f$ and $g$ belong to the polynomial ring $T[x]$ and at least one of the fractional ideals $\mathfrak{m}_f$ and $\mathfrak{m}_g$ is \PMlinkescapetext{regular}. (See also product of finitely generated ideals.)\\
\end{thmplain}
\begin{thmplain}
\,\, (Pahikkala 1982) \, The commutative ring $R$ with non-zero unity is Pr\"ufer ring iff the multiplication rule
$$(a,\,b)(c,\,d) = (ac,\,ad+bc,\,bd)$$
for the integral ideals of $R$ holds whenever at least one of the generators $a$, $b$, $c$ and $d$ is not zero divisor.
\end{thmplain}
The proofs are found in the paper
J. Pahikkala 1982: ``Some formulae for multiplying and inverting ideals''.\, -- {\em Annales universitatis turkuensis} 183. Turun yliopisto (University of Turku).\\
Cf. the entries ``\PMlinkname{multiplication rule gives inverse ideal}{MultiplicationRuleGivesInverseIdeal}'' and ``\PMlinkname{two-generator property}{TwoGeneratorProperty}''.
An additional characterization of Pr\"ufer ring is found here in the entry ``\PMlinkname{least common multiple}{LeastCommonMultiple}'', several other characterizations in [1] (p. 238--239).\\
\textbf{Note.}\, A commutative ring $R$ satisfying the equation (1) for all polynomials $f,\,g$ is called a {\em Gaussian ring.}\, Thus any \PMlinkname{Pr\"ufer domain}{PruferDomain} is always a Gaussian ring, and \PMlinkname{conversely}{Converse}, an integral domain, which is a Gaussian ring, is a Pr\"ufer domain.\, Cf. [2].
\begin{thebibliography}{9}
\bibitem{LM}{\sc M. Larsen \& P. McCarthy:} {\em Multiplicative theory of ideals}.\, Academic Press. New York (1971).
\bibitem{SG}{\sc Sarah Glaz:} ``The weak dimensions of Gaussian rings''. -- {\em Proc. Amer. Math. Soc.} (2005).
\end{thebibliography}