Exercise 32, § 1

Let $A$ be a ring. a) The kernel of $\Phi : A^J \to A^J$ consists of the elements $a$ such that $da_d = 0$ for all $d \in J$. b) Suppose given for every prime number $p$ an endomorphism $\sigma_p$ of $A$ such that $\sigma_p(a) \equiv a^p \mod. pA$ for all $a \in A$. Then the image of $\Phi$ consists of the elements $u$ such that $u_{np} \equiv \sigma_p(u_n) \mod. p^{w_p(np)}A$ for all $n \in J$. This image is a subring of $A^J$ stable under $f_n$ and $v_n$ for all $n \in J$. c) Let $q \in J$. If multiplication by $q$ in $A$ is bijective, then multiplication by $q$ is again bijective in $\mathrm{Ker}(\Phi)$ and in $\mathrm{Im}(\Phi)$.