A note on finitely generated ideals which are locally principal
HTML articles powered by AMS MathViewer
- 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
-
N. Bourbaki, Algèbre commutative. Chaps. 1, 2, Actualités Sci. Indust., no. 1290, Hermann, Paris, 1961. MR 36 #146.
- Shizuo Endo, On flat modules over commutative rings, J. Math. Soc. Japan 14 (1962), 284–291. MR 179226, DOI 10.2969/jmsj/01430284
- Shizuo Endo, On semi-hereditary rings, J. Math. Soc. Japan 13 (1961), 109–119. MR 138664, DOI 10.2969/jmsj/01320109
- Robert W. Gilmer, Multiplicative ideal theory, Queen’s Papers in Pure and Applied Mathematics, No. 12, Queen’s University, Kingston, Ont., 1968. MR 0229624
- William W. Smith, Projective ideals of finite type, Canadian J. Math. 21 (1969), 1057–1061. MR 246864, DOI 10.4153/CJM-1969-116-7
- Wolmer V. Vasconcelos, On finitely generated flat modules, Trans. Amer. Math. Soc. 138 (1969), 505–512. MR 238839, DOI 10.1090/S0002-9947-1969-0238839-5
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 31 (1972), 429-432
- DOI: https://doi.org/10.1090/S0002-9939-1972-0289480-5
- MathSciNet review: 0289480