Defeat of the $\textrm {FP}^ 2\textrm {F}$ conjecture: Huckaba’s example

Abstract:

A commutative ring $R$ is $F{P^2}F$ (resp. FPF) provided that all finitely presented (resp. finitely generated) faithful modules generate the category $\bmod - R$ of all $R$-modules. A conjecture of the author dating to the middle 1970s states that any $F{P^2}F$ ring $R$ has FP-injective classical quotient ring $Q = {Q_{cl}}(R)$. 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 and P. Pillay (Classification of commutative FPF rings, Notas Math., vol. 4, Univ. de Murcia, Murcia, Spain, 1990) that $CF{P^2}F$ 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 $CF{P^2}F$ in the sense that every factor ring of $R$ is $F{P^2}F$.