Punctually free ideals

Abstract

7 Concluding remarks: Let I be a nonzero f.g. PF ideal. We have seen that I is invertible iff it is regular, and I is projective iff rk I is constant on a nhbd of each prime; thus, such an I is invertible iff it is a regular projective ideal. Moreover, the Bourbaki example shows that such an I may be projective of constant rk 1 and still not be invertible. From a local perspective the two notions seem very close, yet their global characterizations appear to be quite different, with invertibility being an ideal-theoretic concept and projectivity a module-theoretic concept. A primary reference for the ideal-theoretic topics presented here is Gilmer's influential book Multiplicative Ideal Theory (now available in three editions [Gil68], [Gil72], [Gil92]); for example, one finds there subject headings for invertible ideas, cancellation ideals, almost Dedekind domains, etc. On the other hand, the notion of projective and its offshoots are best pursued in Bourbaki. My thanks to W. Heinzer for his comments and encouragement. © 2006 Springer Science+Business Media, LLC.