13A99-ProofThatEverySubringOfACyclicRingIsAnIdeal.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{ProofThatEverySubringOfACyclicRingIsAnIdeal}
\pmcreated{2013-03-22 13:30:52}
\pmmodified{2013-03-22 13:30:52}
\pmowner{Wkbj79}{1863}
\pmmodifier{Wkbj79}{1863}
\pmtitle{proof that every subring of a cyclic ring is an ideal}
\pmrecord{9}{34100}
\pmprivacy{1}
\pmauthor{Wkbj79}{1863}
\pmtype{Proof}
\pmcomment{trigger rebuild}
\pmclassification{msc}{13A99}
\pmclassification{msc}{16U99}
\pmrelated{ProofThatEverySubringOfACyclicRingIsACyclicRing}

\endmetadata

\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}

\usepackage{psfrag}
\usepackage{graphicx}
\usepackage{amsthm}
%%\usepackage{xypic}
\begin{document}
\PMlinkescapeword{generator}

The following is a proof that every subring of a cyclic ring is an ideal.

\begin{proof}
Let $R$ be a cyclic ring and $S$ be a subring of $R$.  Then $R$ and $S$ are both cyclic rings.  Let $r$ be a \PMlinkname{generator}{Generator} of the additive group of $R$ and $s$ be a generator of the additive group of $S$.  Then $s \in R$.  Thus, there exists $z \in \mathbb{Z}$ with $s=zr$.

Let $t \in R$ and $u \in S$.  Then $u \in R$.  Since multiplication is commutative in a cyclic ring, $tu=ut$.  Since $t \in R$, there exists $a \in {\mathbb Z}$ with $t=ar$.  Since $u \in S$, there exists $b \in {\mathbb Z}$ with $u=bs$.

Since $R$ is a ring, $r^2 \in R$.  Thus, there exists $k \in {\mathbb Z}$ with $r^2=kr$.  Since $tu=(ar)(bs)=(ar)[b(zr)]=(abz)r^2=(abz)(kr)=(abkz)r=(abk)(zr)=(abk)s \in S$, it follows that $S$ is an ideal of $R$.
\end{proof}

\begin{thebibliography}{9}
\bibitem{buck} Buck, Warren. \emph{\PMlinkexternal{Cyclic Rings}{http://planetmath.org/?op=getobj&from=papers&id=336}}. Charleston, IL: Eastern Illinois University, 2004.

\bibitem{maurer} Maurer, I. Gy. and Vincze, J. ``Despre Inele Ciclece.'' \emph{Studia Universitatis Babe\c{s}-Bolyai. Series Mathematica-Physica}, vol. 9 \#1. Cluj, Romania: Universitatea Babe\c{s}-Bolyai, 1964, pp. 25-27.
\end{thebibliography}
%%%%%
%%%%%
\end{document}