Idempotents in $\mathbb Z_n$

An element $a$ of the ring $(P,+,\cdot)$ is called idempotent if $a^2=a$. An idempotent $a$ is called nontrivial if $a \neq 0$ and $a \neq 1$.

My question concerns idempotents in rings $\mathbb Z_n$, with addition and multiplication modulo $n$, where $n$ is natural number. Obviously when $n$ is a prime number then there is no nontrivial idempotent. If $n$ is nonprime it may happen, for example $n=4, n=9$, that also there is no.

Is it known, in general, for what $n$ there are nontrivial idempotents and what is a form of such idempotents?

As often happens when dealing with $\mathbf{Z}_n$, the Chinese remainder theorem is your friend. If the prime factorization of $n$ is $$ n=\prod_i p_i^{a_i}, $$ then by CRT we have an isomorphism of rings $$ \mathbf{Z}_n\cong\bigoplus_i \mathbf{Z}_{p_i^{a_i}}. $$ Observe that the isomorphism maps the residue class of an integer $m$ (modulo $n$) to a vector with all the components equal to the residue class of $m$ (this time modulo various prime powers): $$ \overline{m}\mapsto(\overline{m},\overline{m},\ldots,\overline{m}). $$ So the residue class of $m$ is an idempotent if and only if it is an idempotent modulo all the prime powers $p_i^{a_i}$.

Let us look at the case of a prime power modulus $p^t$. The congruence $x^2\equiv x\pmod{p^t}$ holds, iff $p^t$ divides $x^2-x=x(x-1)$. Here only one of the factors of, $x$ or $(x-1)$, can be divisible by $p$, so for the product to be divisible by $p^t$ the said factor then has to be divisible by $p^t$. Thus we can conclude that $x\equiv 0,1 \pmod{p^t}$ are the only idempotents modulo $p^t$. Therefore we require that $$ m\equiv 0,1\pmod{p_i^{a_i}} $$ for all $i$. By CRT these congruences are independent for different $i$, so the number of pairwise non-congruent idempotents is equal to $2^\ell$, where $\ell$ is the number of distinct prime factors $p_i$ of $n$.

Hint $ $ Idempotents in $\:\Bbb Z_{ n}\:$ correspond to factorizations of $\:n\:$ into two coprime factors. Namely, if $\:e^2 = e\in\Bbb Z_n\:$ then $\:n\:|\:e(e\!-\!1)\:$ thus $\:n = jk,\ j\:|\:e,\ k\:|\:e\!-\!1,\:$ so $\:(j,k)= 1\:$ by $\:(e,e\!-\!1) = 1.\:$ Conversely if $\:n = jk\:$ for $\:(j,k)= 1,\:$ then by CRT, $\:\Bbb Z_n\cong \Bbb Z_j\times \Bbb Z_k\:$ which has nontrivial idempotents $\:(0,1),\,(1,0).\:$ It is not that difficult to explicitly work out the details of the correspondence. Some integer factorization algorithms search for such nontrivial idempotents.

This correspondence between idempotents and factorizations holds more generally at the structural level - see the Peirce Decomposition. and this answer.