A lambda-ring $R$ is called "special" if it satisfies the $\lambda^i\left(xy\right)=...$ and $\lambda^i\left(\lambda^j\left(x\right)\right)=...$ relations, or, equivalently, if the map $\lambda_T:R\to\Lambda\left(R\right)$ given by $\lambda_T\left(x\right)=\sum\limits_{i=0}^{\infty}\lambda^i\left(x\right)T^i$ (where the $\sum$ sign means addition in $R\left[\left[T\right]\right]$, not addition in $\Lambda\left(R\right)$) is a morphism of lambda-rings. If you are wondering what the hell I am talking about, most likely you belong to the school of algebraists that denote only special lambda-rings as lambda-rings at all.

Anyway, let $A$ and $B$ be two special lambda-rings, and for every $i>0$, let $\Psi_A^i$ and $\Psi_B^i$ be the $i$-th Adams operations on $A$ and $B$, respectively. Let $f:A\to B$ be a ring homomorphism such that $f\circ\Psi_A^i=\Psi_B^i\circ f$ for every $i>0$. Does this yield that $f$ is a lambda-ring homomorphism, i. e. that $f\circ\lambda_A^i=\lambda_B^i\circ f$ for every $i>0$ ?

Note that this is clear if both $A$ and $B$ are torsion-free as additive groups (i. e., none of the elements $1$, $2$, $3$, ... is a zero-divisor in any of the rings $A$ and $B$), but Hazewinkel, in his text Witt vectors, part 1 (Lemma 16.35), claims the same result for the general case. I am writing a list of errata for his text, and I would like to know whether this should be included - well, and I'd like to know the answer anyway, as I am writing some notes on lambda-rings as well.

For the sake of completeness, here is a definition of Adams operations: These are the maps $\Psi^i:R\to R$ for every integer $i>0$ (where $R$ is a special lambda-ring) defined by the equation

$\sum\limits_{i=1}^{\infty} \Psi^i\left(x\right)T^i = -T\frac{d}{dT}\log\left(\lambda_{-T}\left(x\right)\right)$ in the ring $R\left[\left[T\right]\right]$ for every $x\in R$.

Here, even if the term $\log\left(\lambda_{-T}\left(x\right)\right)$ may not make sense (since some of the fractions $\frac{1}{1}$, $\frac{1}{2}$, $\frac{1}{3}$, ... may not exist in $R$), the logarithmic derivative $\frac{d}{dT}\log\left(\lambda_{-T}\left(x\right)\right)$ is defined formally by

$\displaystyle \frac{d}{dT}\log\left(\lambda_{-T}\left(x\right)\right)=\frac{\frac{d}{dT}\lambda_{-T}\left(x\right)}{\lambda_{-T}\left(x\right)}$.