For which fields $K$ and integers $n>1$ does the group ring $K(\mathbb{Z}/n\mathbb{Z})$ have idempotents distinct from $0$ and $1$?



Idempotents in group rings of finite cyclic groups
More generally, when $G$ is a finite group and $\mathbb{K}$ is a field, the group algebra $\mathbb{K}G$ has no non-trivial idempotent if and only if the regular $\mathbb{K}G$-module is indecomposable. If $G$ is non-trivial and the characteristic of $\mathbb{K}$ is either zero or is coprime to $|G|$, then this is never the case, since the augmentation ideal is a direct summand of the regular module in that situation.


If $|G|$ is divisible by the characteristic of $\mathbb{K}$, say $p$, then either $G$ is a $p$-group, in which case the regular $\mathbb{K}G$-module is indecomposable (eg by a theorem of J.A. Green), or else $|G|$ is divisible by some prime $q \neq p,$ in which case $G$ has a conjugacy class of elements of order $q$, and the projective cover of the trivial $\mathbb{K}G$-module has dimension at most $\frac{|G|}{q},$ so that $\mathbb{K}G$ contains an idempotent which is neither zero nor $1$.


Hence when $\mathbb{K}$ has prime characteristic $p$ and $G$ is a finite group, the group algebra $\mathbb{K}G$ has an idempotent other than $0$ or $1$ if and only if $G$ is not a $p$-group, while if $\mathbb{K}$ has characteristic zero and $G$ is a non-trivial finite group, then $\mathbb{K}G$ always has idempotents other than $0$ or $1$.


