Lemma 3

Let A be a ring. There exists a surjective homomorphism of rings $\rho : B \to A$, where B is a ring satisfying the following conditions: p is not a divisor of 0 in B, and there exists an endomorphism $\sigma$ of B such that $\sigma(b) \equiv b^p \mod. p.B$ for all $b \in B$.

It suffices indeed to set $B = \mathbf{Z}[(X_a)_{a \in A}]$, to take for $\sigma$ the endomorphism of B defined by $\sigma(X_a) = X_a^p$ for all $a \in A$, and for $\rho$ the homomorphism of B into A defined by $\rho(X_a) = a$ for all $a \in A$.