Strong prüfer rings and the ring of finite fractions

Abstract

A finite fraction over a commutative ring R is a rational function of the form f=(a nX n +…+ a 0) (b nX n +…+ b 0) for which fb i = a i and a(X), b( X)∈ R[ X]. The collection of all such finite fractions forms a ring Q 0(R) which sits between the total quotient ring of R and the complete ring of quotients of R. We introduce a new type of Prüfer ring, referred to as a Q 0-Prüfer ring and defined as a ring R for which every ring between R and Q 0(R) is integrally closed in Q 0(R) . It is shown that every strong Prüfer ring is a Q 0-Prüfer ring and every Q 0-Prüfer ring is a Prüfer ring. Each converse is shown to be false. However, being a strong Prüfer ring is shown to be equivalent to being a Q 0-Prüfer ring with Q 0(R) having Property A.