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\begin{document}

\title{A Note on Piffles}
\author{A. B. Smith\\Dept. of Pure Mathematics, The University. Sheffield, England}
\date{1966}

\maketitle

\section{Introduction}

Jones in \cite{jones} first defined a Biffle to be a non-definite Boffle and asked if every Biffle was reducible.

Brown in \cite{brown} answered in part this question by defining a Wuffle to be a reducible Biffle and he was then able to show that all Wuffles were reducible.

Green, Smith, and Jones in their review of Brown's paper \cite{green} suggested the name Woffle for any Wuffle other than the nontrivial Wuffle and conjectured that the total number of Woffles would be at least as great as the number so far known to exist. They asked if this conjecture was the strongest possible.

Brown in \cite{brown2} defined a Piffle to be an infinite multi-variable sub-polynormal Woffle which does not satisfy the lower regular $Q$-property. He states, but was unable to prove, that there was at least a finite number of Piffles.

Smith, Jones, Brown, and Green in their collected works \cite{smith} showed that all bi-universal Piffles were strictly descending and conjectured that to prove a stronger result would be harder.

It is this conjecture which motivated the present paper. We aim to show that in

\boldmath 
\[ 
\begin{tikzcd}
R \arrow{r}{\phi} \arrow[swap]{d}{\chi} & S \arrow{d}{\Psi} \\
R/I \arrow{ur}{\psi} \arrow{r}{\Phi} & T
\end{tikzcd}
\quad \mbox{we have that} \,
\psi \mbox{ is a } \left\{ \begin{array}{ll} \mbox{Piffle} 
& \mbox{if } R \mbox{ is polynomial}\\
\mbox{Woffle} & \mbox{if } R 
\mbox{ is Noetherian}\\ \end{array} \right.
\] 

\begin{thebibliography}{9}

\bibitem{brown}
C.D.~Brown: On a paper by A.C.~Jones, {\em Biffle} 24 (1958)

\bibitem{brown2}
T.~Brown, in: {\em A collection of 250 papers on Woffle Theory dedicated to the honor of R.S.Green on his 23rd birthday}, Springer (1961)

\bibitem{green}
H.~Green, P.~Smith, and D.~Jones: Review of a paper by Brown, 
{\em Wuffle Review} 48 (1959)

\bibitem{jones}
A.C.~Jones: A Note on the Theory of Boffles, 
{\em Proceedings of the National Society} 13 (1956)

\bibitem{smith}
T.~Smith, L.~Jones, R.~Brown, and A.~Green:
{\em A short introduction to the classical theory of the Piffle}, Piffle Press (1964)

\end{thebibliography}

\end{document}

