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Punctually free ideals

In: Multiplicative Ideal Theory in Commutative Algebra

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  • Jack Ohm

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.

Suggested Citation

  • Jack Ohm, 2006. "Punctually free ideals," Springer Books, in: James W. Brewer & Sarah Glaz & William J. Heinzer & Bruce M. Olberding (ed.), Multiplicative Ideal Theory in Commutative Algebra, pages 311-330, Springer.
    • Handle: RePEc:spr:sprchp:978-0-387-36717-0_19
    DOI: 10.1007/978-0-387-36717-0_19
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