13A99-AFiniteRingIsCyclicIfAndOnlyItsOrderAndCharacteristicAreEqual.tex

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\pmcanonicalname{AFiniteRingIsCyclicIfAndOnlyItsOrderAndCharacteristicAreEqual}
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\pmtitle{a finite ring is cyclic if and only its order and characteristic are equal}
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\pmtype{Theorem}
\pmcomment{trigger rebuild}
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{\bf \PMlinkescapetext{Lemma}.}  A finite ring is cyclic if and only if its \PMlinkname{order}{OrderRing} and characteristic are equal.  

\begin{proof}
If $R$ is a cyclic ring and $r$ is a \PMlinkname{generator}{Generator} of the additive group of $R$, then $|r|=|R|$.  Since, for every $s \in R$, $|s|$ divides $|R|$, then it follows that $\operatorname{char}~R=|R|$.  Conversely, if $R$ is a finite ring such that $\operatorname{char}~R=|R|$, then the exponent of the additive group of $R$ is also equal to $|R|$.  Thus, there exists $t \in R$ such that $|t|=|R|$.  Since $\langle t \rangle$ is a subgroup of the additive group of $R$ and $|\langle t \rangle |=|t|=|R|$, it follows that $R$ is a cyclic ring.\end{proof}
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