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Community Bot
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I posed the question herehere, but get no answers yet.

Let $R$ be a PID, $M$ be an $R$-module. If $M$ is isomorphic to $r$ copies of cyclic primary module $R/\langle p^s\rangle$ where $p$ is a prime element of $R$, then does $M$ possess the following property?

Given any submodule $N$ of $M$ isomorphic to $R/\langle p^{s_1}\rangle\oplus\cdots\oplus R/\langle p^{s_r}\rangle$, $M/N$ is isomorphic to $R/\langle p^{s-s_1}\rangle\oplus\cdots\oplus R/\langle p^{s-s_r}\rangle$.

I posed the question here, but get no answers yet.

Let $R$ be a PID, $M$ be an $R$-module. If $M$ is isomorphic to $r$ copies of cyclic primary module $R/\langle p^s\rangle$ where $p$ is a prime element of $R$, then does $M$ possess the following property?

Given any submodule $N$ of $M$ isomorphic to $R/\langle p^{s_1}\rangle\oplus\cdots\oplus R/\langle p^{s_r}\rangle$, $M/N$ is isomorphic to $R/\langle p^{s-s_1}\rangle\oplus\cdots\oplus R/\langle p^{s-s_r}\rangle$.

I posed the question here, but get no answers yet.

Let $R$ be a PID, $M$ be an $R$-module. If $M$ is isomorphic to $r$ copies of cyclic primary module $R/\langle p^s\rangle$ where $p$ is a prime element of $R$, then does $M$ possess the following property?

Given any submodule $N$ of $M$ isomorphic to $R/\langle p^{s_1}\rangle\oplus\cdots\oplus R/\langle p^{s_r}\rangle$, $M/N$ is isomorphic to $R/\langle p^{s-s_1}\rangle\oplus\cdots\oplus R/\langle p^{s-s_r}\rangle$.

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Binzhou Xia
Binzhou Xia
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Homocyclic primary module over PID

I posed the question here, but get no answers yet.

Let $R$ be a PID, $M$ be an $R$-module. If $M$ is isomorphic to $r$ copies of cyclic primary module $R/\langle p^s\rangle$ where $p$ is a prime element of $R$, then does $M$ possess the following property?

Given any submodule $N$ of $M$ isomorphic to $R/\langle p^{s_1}\rangle\oplus\cdots\oplus R/\langle p^{s_r}\rangle$, $M/N$ is isomorphic to $R/\langle p^{s-s_1}\rangle\oplus\cdots\oplus R/\langle p^{s-s_r}\rangle$.