Abstract
Let D be an integral domain and w be the so-called w-operation on D. We define D to be a w-FF domain if every w-flat w-ideal of D is of w-finite type. This paper presents some properties of w-FF domains and related domains. Among other things, we study the w-FF property in the polynomial extension, the t-Nagata ring and the pullback construction.
Highlights
In order to study some questions of divisibility and finite generation, Sally and Vasconcelos first studied an integral domain in which every flat ideal is finitely generated [1]
An FF domain is an integral domain in which each nonzero flat ideal is invertible, and a weakly FF domain can be regarded as an integral domain in which each faithfully flat ideal is invertible
It is well known that an ideal I of D is flat if and only if IDP is flat for all prime ideals P of D [10] (Proposition 3.10); so a nonzero flat ideal is w-flat
Summary
As a weaker version, the authors defined a weakly FF property on D such that each faithfully flat ideal of D is finitely generated. In [4], the authors introduced and studied the notion of a w-LPI domain; that is, an integral domain in which every nonzero w-locally principal ideal is w-invertible. The notion of w-LPI domains coincides with that of weakly w-FF domains because an ideal is w-faithfully flat if and only if it is w-locally principal [6] (Theorem 2.7). We give several examples which represent relationships among integral domains related to w-FF domains (Example 1)
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