A note on ordinal numbers and rings of formal power series

Abstract

In [3], Simpson has shown that over R C A o, for any or all countable fields K, a formal version of Hilbert basis theorem is equivalent to the assertion that the ordinal number co ~~ is well ordered. It is well-known that there is a basis theorem for rings of formal power series whose statement is "Let R be a commutative ring whose all ideals are finitely generated. Then, all ideals of the commutative ring of formal power series with coefficients from R are also finitely generated." In this paper we establish that co ~ also "measures" the "intrinsic logical strength" of a version of this assertion formalised in second order arithmetic and in which the ring of coefficients can be any countable field. We recall that R C A o is a subsystem of second order arithmetic; it consists of all the familiar ordered semiring axioms for (N, + , .,0, 1, <), together with the Z ~ induction scheme and the A ~ comprehension scheme (for details, see [1]). Within R C A o we can prove that for any countable commutative ring R and n E N there exists a countable commutative ring R[x l , 322, . . . , Xn] (the ring of polynomials in n commuting indeterminates xl, x2, . . . , x~ over R) consisting of 0 and all (G6del numbers of) expressions of the form: