HOW TO ELIMINATE QUANTIFIERS IN THE ELEMENTARY THEORY OF p-RINGS by BUREHARD MOLZAN in Berlin (G.D.R.)l) 0. Int,roduction Let p be an arbitrary prime. By a p-ring we understand a commutative ring with unit, any element of which satisfies XP = x and px =df x + . . . + x = 0. Let p times LR = (+, *, 0 , l ) be the language of ring theory and CR(p) the elementary theory of the class of all p-rings in LR.
Whas we want to do is the following: To construct a (recursive) set of LR-formulas, say S, such that any formula of LR in CR(p) is equivalent to some Boolean combination of members of S and atomic LR-formulas. Furthermore each member of S should express some algebraically relevant property. 1.e. we want to state the essential facts on p-rings expressible in LR.
First we will use a representastion theorem of &ENS and KAPLANSKY t o obtain a convenient treatment of terms in CR(p). Then we will apply ideas of KOZEN to define a set of formulas possessing the required properties. Finally a description of all complete extensions of CR(p) will be given analogously to the ERsHov-classification of all complete theories of Boolean algebras. Beside we will prove the decidability of CR(p) and some facts on atomless p-rings.