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On the Criteria of D.D. Anderson for Invertible and Flat Ideals

Published online by Cambridge University Press:  20 November 2018

David E. Dobbs
David E. Dobbs*
Affiliation:
Department of Mathematics, University of TennesseeKnoxville, Tennessee 37996, U.S.A.
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Abstract

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Let R be an integral domain. It is proved that if a nonzero ideal I of R can be generated by n < ∞ elements, then I is invertible (i.e., flat) if and only if I(∩ Rai) = ∩ Iai for all { a1, . . ., a n﹜ ⊂ I. The article's main focus is on torsion-free R-modules E which are LCM-stable in the sense that E(Ra ∩ Rb) = Ea ∩ Eb for all a, b ∈ R. By means of linear relations, LCM-stableness is shown to be equivalent to a weak aspect of flatness. Consequently, if each finitely generated ideal of R may be 2-generated, then each LCM-stable R-module is flat. Finally, LCM-stableness of maximal ideals serves to characterize Prüfer domains, Dedekind domains, principal ideal domains, and Bézout domains amongst suitably larger classes of integral domains.

Keywords

primary 13C1113F05secondary 13G0513F1513E0518G15
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 29 , Issue 1 , 01 March 1986 , pp. 25 - 32
Copyright
Copyright © Canadian Mathematical Society 1986

References

1. Anderson, D. D., On the ideal equation I(B ∩ C) = IB ∩ IC, Canad. Math. Bull. 26 (1983), pp. 331–332.Google Scholar
2. Bourbaki, N., Commutative algebra, Addison-Wesley, Reading, Mass. 1972.Google Scholar
3. Chase, S. U., Direct products of modules, Trans. Amer. Math. Soc. 97 (1960), pp. 457–473.Google Scholar
4. Dobbs, D. E., On flat finitely generated ideals, Bull. Austral. Math. Soc. 21 (1980), pp. 131–135.Google Scholar
5. Dobbs, D. E. and Papick, I. J., When is D + M coherent?, Proc. Amer. Math. Soc. 56 (1976), pp. 51–54.Google Scholar
6. Eakin, P. and Sathaye, A., Prestable ideals, J. Algebra 41 (1976), pp. 439–454.Google Scholar
7. Endo, S., On flat modules over commutative rings, J. Math. Soc. Japan 14 (1962), pp. 284–291.Google Scholar
8. Gilmer, R., Multiplicative ideal theory, Dekker, New York, 1972.Google Scholar
9. Gilmer, R., Finite element factorization in group rings, Lecture Notes in Pure and Appl. Math., 7, Dekker, New York, 1974.Google Scholar
10. Jensen, Chr. U. On characterizations of Prüfer rings, Math. Scand. 13 (1963), pp. 90—98.Google Scholar
11. Jensen, Chr. U., A remark on flat and projective modules, Canad. J. Math. 18 (1966), pp. 943—949.Google Scholar
12. Kaplansky, I., Commutative rings, rev. ed., Univ. of Chicago Press, Chicago, 1974.Google Scholar
13. Richman, F., Generalized quotient rings, Proc. Amer. Math. Soc. 16 (1965), pp. 794—799.Google Scholar
14. Sally, J. D. and Vasconcelos, W. V., Flat ideals, I, Comm. In Algebra 3 (1975), pp. 531–543.Google Scholar
15. Sheldon, P., Prime ideals in GCD-domains, Canad. J. Math. 26 (1974), pp. 98–107.Google Scholar
16. Uda, H., LCM-stableness in ring extensions, Hiroshima Math. J. 13 (1983), pp. 357—377.Google Scholar
17. Vasconcelos, W. V., The local rings of global dimension two, Proc. Amer. Math. Soc. 35 (1972), pp. 381–386.Google Scholar
18. Vasconcelos, W. V., Divisor theory in module categories, Math. Studies, 14, North-Holland, Amsterdam, 1974.Google Scholar