Let $p^r \mathbb{Z} (r\geq 2)$ be a submodule of $\mathbb{Z}$. Is it a $p$-primary subsmodule of $\mathbb{Z}$?
Please guide me with a answer. Thank you for your kindness.
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2How is $p$-primary defined? – Tobias Kildetoft Aug 02 '13 at 08:15
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$\bf Z$-modules are just abelian groups; wouldn't the $p$-primary component be the subgroup of all elements with $p$-power torsion? The group $\bf Z$ has no torsion elements. – anon Aug 02 '13 at 08:15
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What's a $,p$-primary module? The comment by anon makes one wonder how you define that... – DonAntonio Aug 02 '13 at 08:22
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@al-Hwarizmi But that definition does not have any obvious way to be changed into $p$-primary, so it is not clear that they are related. – Tobias Kildetoft Aug 02 '13 at 08:30
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Definition: Let $M$ be an $R$-module. A proper submodule $N$ of $M$ is called primary if whenever for every $a\in R$ and $m\in M$ such that $am\in N$, then either $a\in \sqrt{(N :_R M)}$ or $m\in N$. In this case, if $\sqrt{(N :_R M)}= p$ such taht $p$ is a prime ideal of $R$, then $N$ is called a $p$-primary submodule of $M$. – Saniya Aug 02 '13 at 08:34