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Let $R$ be a commutative ring, $W(R)$ is a ring of Witt vectors over $R$. Can you give an example of a ring $R$ such that $W(R)$ has $p$-torsion?

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    $\begingroup$ For example, $R = \mathbb{F}_p\left[x\right] / \left(x^p\right)$. Then, if you realize the Witt vectors $W\left(R\right)$ as power series with constant term $1$ over $R$ (with multiplication of power series as addition), then the Witt vector $1 + tx$ becomes $0$ when multiplied by $p$. $\endgroup$
    – darij grinberg
    Commented Jun 16, 2018 at 22:03
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    $\begingroup$ You could also use $R = \mathbf Z/p^2\mathbf Z$. Here $p^i = 0$ for all $i \geq 2$ and $a = (p,p,p,\ldots)$ is a nonzero element of $W(R)$ with $pa = 0$. $\endgroup$
    – KConrad
    Commented Jun 17, 2018 at 1:15

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