A note on ordinal numbers and rings of formal power series (Q1337497)

In ``Ordinal numbers and the Hilbert basis theorem'' [J. Symb. Log. 53, No. 3, 961-974 (1988; Zbl 0661.03046)], \textit{S. G. Simpson} has shown that over \(\text{RCA}_ 0\), for any or all countable fields \(K\), a formal version of Hilbert basis theorem is equivalent to the assertion that the ordinal number \(\omega^ \omega\) 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 all of whose 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 \(\omega^ \omega\) also ``measures'' the ``intrinsic logical strength'' of a version of this assertion formalized in second-order arithmetic and in which the ring of coefficients can be any countable field.