A note on finitely generated ideals which are locally principal

Brewer, James W.; Rutter, Edgar A.

doi:10.1090/S0002-9939-1972-0289480-5

A note on finitely generated ideals which are locally principal
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by James W. Brewer and Edgar A. Rutter PDF

Proc. Amer. Math. Soc. 31 (1972), 429-432 Request permission

Abstract:

Let $R$ be a commutative ring with identity $1 \ne 0$ and let $A$ be a nonzero ideal of $R$. A problem of current interest is to relate the notions of “projective ideal", “flat ideal” and “multiplication ideal". In this note we prove two results which show that the maximal ideals containing the annihilator of $A$ can play an important role in determining the relationship between these concepts. As a consequence we are able to prove that a finitely generated multiplication ideal in a semi-quasi-local ring is principal, that a finitely generated flat ideal having only a finite number of minimal prime divisors is projective and that for Noetherian rings or semihereditary rings, finitely generated multiplication ideals with zero annihilator are invertible.

References

Algèbre commutative

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Additional Information

Journal: Proc. Amer. Math. Soc. 31 (1972), 429-432
DOI: https://doi.org/10.1090/S0002-9939-1972-0289480-5
MathSciNet review: 0289480