Krull rings, Prüfer \(v\)-multiplication rings and the ring of finite fractions
This paper deals with extending the notions of Krull domains and PvMDs to rings with zero divisors. Two of the problems to address involve characterizing when a Nagata ring will be a Krull ring and when it will be a PvMR. For both problems, the characterizations entail considering how the ring in question is placed in its associated ring of finite fractions. Reviewer's remark. The notion of the ring of finite fractions was introduced by the author in previous papers [J. Pure Appl. Algebra 61, 49--52 (1989; Zbl 0697.13003) and in: Commutative ring theory (Fès, 1992), Lecture Notes Pure Appl. Math. 153, 181--191 (1994; Zbl 0865.13003)]. The author found some interesting applications of this ring, which has a very significant role in the paper under review. For these reasons, and also because the reviewer finds the ``finite fractions'' interesting and also rather curious, the definition of the ring of finite fractions is given here. Let \(R\) be a commutative ring with identity. As usual, we use \(R[X]\) to denote the polynomial ring over \(R\). We define the ring of finite fractions over \(R\) as the subring of the total quotient ring of \(R[X]\) consisting of the following fractions: \(f=b(X)/a(X)\), where \(a(X)=\sum a_iZ^i\), \(b(X)=\sum b_jX^j\in R[X]\), with \(\deg(b(X))\leq \deg(a(X))\) such that \(fa_i=b_i\) for each coefficient \(a_i\) of \(a(X)\).
scientific article; zbMATH DE number 5001008
scientific article