STRONG PRUFER-RINGS AND THE RING OF FINITE FRACTIONS

A finite fraction over a commutative ring R is a rational function of the form f = (a(n)X(n) + ... + a0)/(b(n)X(n) + ... + b0) for which fb(i) = a(i) and a(X), b(X) is-an-element-of R[X]. The collection of all such finite fractions forms a ring Q0(R) which sits between the total quotient ring of R and the complete ring of quotients of R. We introduce a new type of Prufer ring, referred to as a Q0-Prufer ring and defined as a ring R for which every ring between R and Q0(R) is integrally closed in Q0(R). It is shown that every strong Prufer ring is a Q0-Prufer ring and every Q0-Prufer ring is a Prufer ring. Each converse is shown to be false. However, being a strong Prufer ring is shown to be equivalent to being a Q0-Prufer ring with Q0(R) having Property A.