Rings for which certain flat modules are projective

The finiteness of 𝐼 when 𝑅[𝑋]/𝐼 is 𝑅-projective

“…That (1) implies (2) is well known and that (2) implies (3) is obvious. Thus, we need only prove that (3) implies (1).…”

“…A is a pure ideal of R if R/A is a flat /{-module. (See [2,Proposition 3.4] for several equivalent formulations; the terminology there being »-ideal. Pure ideal seems to us more appealing, since it calls to mind the fact that 0-*A-*R is a pure sequence of Ä-modules- (1) I =fR[X], f a monk polynomial in R[X].…”

“…But pure ideals need not be principal. In fact, the rings R for which each pure ideal is principal have been much studied and termed y4(0)-rings in [2]. This is a large class of rings including all integral domains, all Noetherian rings and all rings having only a finite number of maximal ideals.…”