Geodesic
On the Criteria of D.D. Anderson for Invertible and Flat Ideals
Canadian mathematical bulletin, Tome 29 (1986) no. 1, pp. 25-32

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-1986-004-4
Mots-clés : primary 13C11, 13F05, secondary 13G05, 13F15, 13E05, 18G15 
Dobbs, David E. On the Criteria of D.D. Anderson for Invertible and Flat Ideals. Canadian mathematical bulletin, Tome 29 (1986) no. 1, pp. 25-32. doi: 10.4153/CMB-1986-004-4
@article{10_4153_CMB_1986_004_4,
     author = {Dobbs, David E.},
     title = {On the {Criteria} of {D.D.} {Anderson} for {Invertible} and {Flat} {Ideals}},
     journal = {Canadian mathematical bulletin},
     pages = {25--32},
     year = {1986},
     volume = {29},
     number = {1},
     doi = {10.4153/CMB-1986-004-4},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1986-004-4/}
}
TY  - JOUR
AU  - Dobbs, David E.
TI  - On the Criteria of D.D. Anderson for Invertible and Flat Ideals
JO  - Canadian mathematical bulletin
PY  - 1986
SP  - 25
EP  - 32
VL  - 29
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1986-004-4/
DO  - 10.4153/CMB-1986-004-4
ID  - 10_4153_CMB_1986_004_4
ER  - 
%0 Journal Article
%A Dobbs, David E.
%T On the Criteria of D.D. Anderson for Invertible and Flat Ideals
%J Canadian mathematical bulletin
%D 1986
%P 25-32
%V 29
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-1986-004-4/
%R 10.4153/CMB-1986-004-4
%F 10_4153_CMB_1986_004_4

[1] 1. Anderson, D. D., On the ideal equation I(B ∩ C) = IB ∩ IC, Canad. Math. Bull. 26 (1983), pp. 331–332. Google Scholar

[2] 2. Bourbaki, N., Commutative algebra, Addison-Wesley, Reading, Mass. 1972. Google Scholar

[3] 3. Chase, S. U., Direct products of modules, Trans. Amer. Math. Soc. 97 (1960), pp. 457–473. Google Scholar

[4] 4. Dobbs, D. E., On flat finitely generated ideals, Bull. Austral. Math. Soc. 21 (1980), pp. 131–135. Google Scholar

[5] 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] 6. Eakin, P. and Sathaye, A., Prestable ideals, J. Algebra 41 (1976), pp. 439–454. Google Scholar

[7] 7. Endo, S., On flat modules over commutative rings, J. Math. Soc. Japan 14 (1962), pp. 284–291. Google Scholar

[8] 8. Gilmer, R., Multiplicative ideal theory, Dekker, New York, 1972. Google Scholar

[9] 9. Gilmer, R., Finite element factorization in group rings, Lecture Notes in Pure and Appl. Math., 7, Dekker, New York, 1974. Google Scholar

[10] 10. Jensen, Chr. U. On characterizations of Prüfer rings, Math. Scand. 13 (1963), pp. 90—98. Google Scholar

[11] 11. Jensen, Chr. U., A remark on flat and projective modules, Canad. J. Math. 18 (1966), pp. 943—949. Google Scholar

[12] 12. Kaplansky, I., Commutative rings, rev. ed., Univ. of Chicago Press, Chicago, 1974. Google Scholar

[13] 13. Richman, F., Generalized quotient rings, Proc. Amer. Math. Soc. 16 (1965), pp. 794—799. Google Scholar

[14] 14. Sally, J. D. and Vasconcelos, W. V., Flat ideals, I, Comm. In Algebra 3 (1975), pp. 531–543. Google Scholar

[15] 15. Sheldon, P., Prime ideals in GCD-domains, Canad. J. Math. 26 (1974), pp. 98–107. Google Scholar

[16] 16. Uda, H., LCM-stableness in ring extensions, Hiroshima Math. J. 13 (1983), pp. 357—377. Google Scholar

[17] 17. Vasconcelos, W. V., The local rings of global dimension two, Proc. Amer. Math. Soc. 35 (1972), pp. 381–386. Google Scholar

[18] 18. Vasconcelos, W. V., Divisor theory in module categories, Math. Studies, 14, North-Holland, Amsterdam, 1974. Google Scholar

Cité par Sources :