ON PRUFER NON-D-RINGS

[摘要] Call a commutative integral domain R with unity a non-D-ring if there exists a nonconstant polynomial f(x) over R such that f(a) is a unit in R for all a in R. This paper is concerned with Prufer non-D-rings which have a monic unit valued polynomial. First, we prove that the ideal class group of such a ring is torsion. Secondly, we show that such rings can be precisely characterized as intersections of non-D valuations domains which share a common monic unit valued polynomial. Finally, let R be a Prufer non-D-ring with a monic unit valued polynomial and with field of quotients K. Let W(R) be the set of all rational functions f(x)/g(x) in K(x) such that f(a)/g(a) is in R for all a in R. Also, for I an ideal of W(R) and a in R, let I(a) be the ideal of R formed by evaluating each element of I at a. We prove that if I and J are finitely generated ideals of W(R) such that I(a) = J(a) for all a in R then I = J (i.e., W(R) has the strong Skolem property).

[发布日期] 1994-10-21