13A99-ProofThatEverySubringOfACyclicRingIsACyclicRing.tex

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\pmtitle{proof that every subring of a cyclic ring is a cyclic ring}
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\pmrelated{ProofThatAllSubgroupsOfACyclicGroupAreCyclic}
\pmrelated{ProofThatEverySubringOfACyclicRingIsAnIdeal}

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\PMlinkescapeword{cyclic}

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

\begin{proof}
Let $R$ be a cyclic ring and $S$ be a subring of $R$.  Then the additive group of $S$ is a subgroup of the additive group of $R$.  By definition of cyclic group, the additive group of $R$ is \PMlinkname{cyclic}{CyclicGroup}.  Thus, the additive group of $S$ is cyclic.  It follows that $S$ is a cyclic ring.
\end{proof}
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