I was asking to myself the following question. Consider a $p$-adically complete and separated topological algebra $R$ over $\mathbb{Z}_{p}$. As $\mathbb{Z}_{p}$ is a domain, we know that the $\mathbb{Z}_{p}$-torsion elements of $R$ form a submodule. Is it possible to prove that this torsion is a closed submodule of $R$? I'm interested in the possibility of saying that the quotient of $R$ over its torsion is again $p$-adically complete, but in order to do this, I think I need that the torsion submodule is closed. Have you got any idea? I tried to prove it by hands but I didn't reach anything useful. Thank you very much!
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Take $$ R=\prod_{n=1}^\infty \mathbb{Z}/p^n\mathbb{Z}, $$ with the product topology. Every non-empty open set in $R$ contains an element that is $0$ in all but finitely many factors, and this element is $\mathbb{Z}_p$-torsion. This means the torsion is dense, and since $1\in R$ is not torsion, the torsion submodule is not closed.
answered Mar 29, 2016 at 21:24
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3$\begingroup$ Isn't the $p$-adic topology on $R$ strictly finer than the product topology? One could rather consider the $p$-adic completion $\widehat{S}$ of $S = \bigoplus_{n=1}^\infty \mathbb{Z} / p^n \mathbb{Z}$. Indeed, $\widehat{S}$ is $p$-adically complete with dense $p$-torsion (it is isomorphic to the submodule of $R$ consisting of those elements whose components have $p$-valuation going to infinity). $\endgroup$ Commented Feb 7, 2017 at 10:16