Independence of the base in Kronecker-style divisor theory (Q1803509)

The author obtains a positive answer to a neutral question from Kronecker's divisor theory, recently reconsidered by \textit{H. M. Edwards} in the monograph ``Divisor theory'' (1990; Zbl 0689.12001)]. He proves that if \(A\) is the integral closure of two different unique factorization domains, \(R\) and \(S\), and \(f\) and \(g\) are polynomials with coefficients in the field of fractions of \(A\), then the polynomials \(g\) divided by the divisor of \(f\) in the theory based on \(R\) are the same as those in the theory based on \(S\). For proving this there is described a necessary and sufficient condition for that a divisor \([f]\) divides \(g\). The condition is expressed using the concept of denominator of a polynomial. The paper also includes a suggestive geometrical motivation of the question discussed.