Decidability of the class of all the rings : A problem of Ax

Abstract We prove that the class of all rings $\mathbb {Z}/m\mathbb {Z}$ for $m>1$ is decidable. This gives a positive solution to problem Ax asked in his celebrated 1968 paper on elementary theory finite fields [1, Problem 5, p. 270]. In our proof, we reduce decidability ring adeles {A}_{\mathbb {Q}}$ {Q}$ .