Let $R$ be a commutative ring with identity and let $f \in R[x]$. There are well known characterizations for $f$ to be a nilpotent element of $R[x]$ or to have a multiplicative inverse in $R[x]$. Is there any characterization for idempotent elements in $R[x]$ ?



Idempotent polynomials
Let $f = a\_0 + a\_1x + ... + a\_nx^n$ be idempotent. Then $a\_0^2 = a\_0$. Also 
$a\_0a\_1 + a\_1a\_0 = a\_1$. Multiply by $a\_0$ to get $a\_0a\_1 = 0$ which 
means that $a\_1 = 0$ and by induction it is easy to show that $a\_2 = ... = a\_n = 0$ 
Therefore **$f$ is idempotent iff its constant term is idempotent and other coefficients are 
zero.** 


Note that this is not true if we drop the commutativity condition. For example 
consider the polynomial $f(x) = \begin{pmatrix}
1 & 0 \\
0 & 0 \\
\end{pmatrix} + \begin{pmatrix}
0 & 1 \\
0 & 0 \\
\end{pmatrix}x $ in $M\_2(\Bbb{R})[x]$ which is clearly an idempotent polynomial. 


