Defeat of the FP 2 F Conjecture: Huckaba's Example

Carl Faith
1992 Proceedings of the American Mathematical Society  
A commutative ring R is FP2F (resp. FPF) provided that all finitely presented (resp. finitely generated) faithful modules generate the category mod-R of all R-modules. A conjecture of the author dating to the middle 1970s states that any FP2F ring R has FP-injective classical quotient ring Q = QciiR) ■ It was shown by the author (Injective quotient rings. II, Lecture Notes in Pure and Appl. Math., vol. 72, Dekker, New York, 1982, pp. 71-105) that FPF rings R have injective Q and by the author
more » ... and by the author and P. Pillay (Classification of commutative FPF rings, Notas Math., vol. 4, Univ. de Murcia, Murcia, Spain, 1990) that CFP2F local rings (defined below) have FP-injective Q . The counterexample is a difficult example due to Huckaba of a strongly Prüfer ring without "Property A." (A ring with Property A was labelled a McCoy ring by the author.) This counterexample is CFP2F in the sense that every factor ring of R is FP2F . Theorem. (I) A ring R is CFP2F iff (2) Rm is a valuation ring (= VR) for each maximal ideal M. Proof. In [Fl] (also [FP]) CFP2F is characterized by the statement: R is locally a VR , i.e., Rp is a VR for all prime ideals P . However, a look at the proof in [Fl, Corollary 5E, p. 176] establishes the equivalence of (1) and (2). A ring R is said to be McCoy (see [F2]) or have Property A (see [H]) provided that every finitely generated dense (= faithful) ideal is regular, i.e., contains a regular element. This is equivalent to the statement that Q = Qc¡iR) is McCoy, i.e., that Q is the only finitely generated dense ideal. A sufficient condition for R to be McCoy is for Q to be FP-injective. (Every FP-injective ring Q has the property that finitely generated ideals are annihilator ideals.) A ring R is Prüfer if every finitely generated regular ideal is invertible (see
doi:10.2307/2159286 fatcat:bsnejppxcvhwnm3wwqbkg22cgi