Let $R$ be a ring, let $I$ be an ideal of $R$, and let $u\in R$ be idempotent modulo $I$ (that is, $u^{2}-u\in I$). Then $u$ can be lifted to an idempotent in $R$ in case there is an idempotent $e$ in $R$ with $e-u \in I$.
I want to show that :
Let $n > 1$ in $\mathbb{N}$. Prove that if $n$ is not a power of a prime, then there exist idempotents modulo $n\mathbb{Z}$ in $\mathbb{Z}$ that cannot be lifted to idempotents in $\mathbb{Z}$