# Retraction given by a principal ideal

Let $R$ be a commutative ring. Let $f : R \to A$ be a smooth morphism of relative dimension 1. Suppose $f$ admits a retraction $r : A \to R$ i.e $r f = Id_R$. Is it true that the kernel of $r$ is a principal ideal ?

-

Let $R$ be an algebraically closed field, let $A$ be the ring of regular functions of a smooth affine curve $C$, then any closed point of $C$ defines a section (=rectraction), but the corresponding maximal ideal is not principal in general because otherwise $A$ would be principal. Concrete example: $C=$ a projective smooth curve of positive genus minus one point. Then no maximal ideal of $A$ is principal.
@DamienL: this is true at the fibers of $\mathrm{Spec}(A)\to\mathrm{Spec}(R)$, then use Nakayama. – user18119 Feb 7 '13 at 16:24