Inductive rings and fields (Q584246)

A ring with unity is said to be inductive iff it satisfies the induction principle, i.e. for every first-order formula F(x) of \(\{0,1,+,\cdot \},\) \([F(0)\wedge (\forall x F(x)\to F(x+1))]\to \forall x F(x).\) (Notice that \({\mathbb{Z}}\) is not inductive, because \({\mathbb{N}}\) is definable as \(\{x^ 2+y^ 2+z^ 2+t^ 2;\quad x,y,z,t\in {\mathbb{Z}}\}!)\) Quotients of \({\mathbb{Z}}\) and algebraically closed fields of characteristic 0 are inductive. The author conjectures that the class of inductive rings is the class generated by these two kinds of structures. He also lists certain noninductive rings. One could regret that the treatment is not more systematic. For example, no number field is inductive, because of the definability of \({\mathbb{Q}}\). Also the two remarks (in the introduction and in 3-2) about the inductivity of algebraically closed fields of characteristic 0 and fields with quantifier elimination are essentially the same, as we know that q.e. fields of characteristic 0 are exactly the algebraically closed fields. But above all, Theorem 1-5, as stated, seems to be incorrect: some power series rings are inductive, for example \({\mathbb{C}}[[X]]\). Note that this ring, as well as \({\mathbb{Z}}_ p\) also is a counter-example to the conjecture mentioned above.