Idempotents in $\mathbf{CRing}$

Question

I'm not able to find an example of an idempotent morphism different from an identity in the category of commutative rings with unity (an idempotent, as a morphism in that category, must preserve 1, right?). Any example of that?

Answer 1

The homomorphism $\mathbf{Z}[x] \to \mathbf{Z}[x]$ given by $f(x) \mapsto f(7)$.

Answer 2

Let $k$ be a field and $R=k[X]/(X^2)$, and consider the unique map of $K$-algebras $\phi:R\to R$ such that $\phi(X)=0$.

Answer 3

Take any idempotent $k$-linear map $V \to V$ on a $k$-module $V$. This induces an idempotent $k$-algebra homomorphism on the symmetric algebra $\mathrm{Sym}(V) \to \mathrm{Sym}(V)$, which is the identity iff $V \to V$ is. We may even take $0 : V \to V$.

For example, if $V$ is free of rank $d$, this gives $k[T_1,\dotsc,T_d] \to k[T_1,\dotsc,T_d],~T_i \mapsto 0$. Another example is $k[T_1,T_2] \to k[T_1,T_2]$, $T_1 \mapsto \lambda T_2$, $T_2 \mapsto T_2$ for $\lambda \in k$, which is induced by the idempotent matrix $\begin{pmatrix}0 & 0 \\ \lambda & 1 \end{pmatrix}$.