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Associated Prime Ideals in Non-Noetherian Rings

Published online by Cambridge University Press:  20 November 2018

Juana Iroz  and
David E. Rush
Juana Iroz
Affiliation:
University of California, Riverside, California
David E. Rush
Affiliation:
University of California, Riverside, California
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The theory of associated prime ideals is one of the most basic notions in the study of modules over commutative Noetherian rings. For modules over non-Noetherian rings however, the classical associated primes are not so useful and in fact do not exist for some modules M. In [4] [22] a prime ideal P of a ring R is said to be attached to an R-module M if for each finite subset I of P there exists m ∊ M such that I ⊆ annR(m) ⊆ P. In [4] the attached primes were compared to the associated primes and the results of [4], [22], [23], [24] show that the attached primes are a useful alternative in non-Noetherian rings to associated primes. Several other methods of associating a set of prime ideals to a module M over a non-Noetherian ring have proven very useful in the past. The most common of these is the set Assf(M) of weak Bourbaki primes of M [2, pp. 289-290].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 2 , 01 April 1984 , pp. 344 - 360
Copyright
Copyright © Canadian Mathematical Society 1984

References

1. Banaschewki, B., On the components of ideals in commutative rings, Arch. Math 12 (1961), 22–29.Google Scholar
2. Bourbaki, N., Commutative algebra (Addison Wesley, Reading, Mass., 1972).Google Scholar
3. Dobbs, D., Lying-over pairs of commutative rings. Can. J. Math. 33 (1981), 454–475.Google Scholar
4. Dutton, P., Prime ideals attached to a module, Quart. J. Math. Oxford 29 (1978), 403–413.Google Scholar
5. Heinzer, W., and Ohm, J., Locally Noetherian commutative rings, Trans. Amer. Math. Soc. 755 (1971), 273–284.Google Scholar
6. Heitmann, R., Prime divisors and flat extensions, J. of Alg. 74 (1982), 293–301.Google Scholar
7. Hochster, M., Grade sensitive modules and perfect modules, Proc. London Math. Soc. 29 (1979), 91–100.Google Scholar
8. Krull, W., Idealtherie in ringen ohne Endlichkeitshedingung, Math. Ann. 101 (1929), 729–744.Google Scholar
9. Kuntz, R. A., Associated prime divisors in the sense of Krull, Can. J. Math. 24 (1972), 808–818.Google Scholar
10. Kuntz, R. A., Associated prime divisors in the sense of Noether, Bol: Soc. Math. Mexicana 18 (1973), 82–88.Google Scholar
11. Lazard, D., Antour de la platitude, Bull. Soc. Math. France 97 (1969), 81–128.Google Scholar
12. MacDonald, I. G., Secondary representation of modules over a commutative ring, Symp. Math 11 (1973), 23–43.Google Scholar
13. MacDonald, I. G. and Sharp, R., An elementary proof of the non-vanishing of certain local cohomology modules, Quart. J. Math., Oxford 23 (1972), 197–204.Google Scholar
14. McDowell, K., A codimension theorem for pseudo-Noetherian rings, Trans. Amer. Math. Soc 274 (1975), 179–185.Google Scholar
15. Merker, J., Pseudo-Noetherian rings, Can. Math. Bull. 19 (1976), 77–84.Google Scholar
16. McDowell, K., Commutative coherent rings, Mathematical Report No. 66, Department of Mathematics, McMaster University, Hamilton, Ontario (1974).Google Scholar
17. McDowell, K., Associated prime divisors and finitely presented modules, unpublished manuscript, Wilfrid Laurier University, Waterloo, Ontario (1975).Google Scholar
18. Merker, J., Etude axiomatic des idéaux premiers associes, C. R. Acad. Sc. Paris 269 (1969), 52–54.Google Scholar
19. Merker, J., Etude axiomatic des idéaux premiers associes, C.R. Acad. Sc. Paris 269 (1969), 117–119.Google Scholar
20. Nagata, M., Local rings (Robert E. Krieger, New York, New York, 1975).Google Scholar
21. Northcott, D. G., A generalization of a theorem on the content of polynomials, Proc. Cambridge Philos. Soc. 55 (1959), 282–288.Google Scholar
22. Northcott, D. G., Finite free resolutions (Cambridge University Press, Cambridge, 1976).CrossRefGoogle Scholar
23. Northcott, D. G., Projective ideals and MacRae's invariant, J. London Math. Soc. 24 (1981), 211–226.Google Scholar
24. Northcott, D. G., Remarks on the theory of attached prime ideals, Quart. J. Math. Oxford 33 (1982), 239–245.Google Scholar
25. Ooishi, A., Matlis duality and the width of a module, Hiroshima Math. J. 6 (1976), 573–587.Google Scholar
26. Rush, D. E., The G-function of MacRae, Can. J. Math. 26 (1974), 854–865.Google Scholar
27. Rush, D. E., Big Cohen Macauley modules, Ill. J. of Math. 24 (1980), 606–611.Google Scholar
28. Rush, D. E., Picard groups in abelian groups rings, J. of Pure and Applied Alg. 26 (1982).Google Scholar
29. Sharp, R. Y., Some results on the vanishing of local cohomology modules, Proc. London Math. Soc. 30 (1975), 177–195.Google Scholar
30. Rush, D. E., Secondary representations for injective modules over commutative Noetherian rings, Proc. Edinburgh Math. Soc. 20 (1976), 143–151.Google Scholar
31. Rush, D. E., On the attached prime ideals of certain Artinian local cohomology modules, Proc. Edinburgh Math. Soc. 24 (1981), 9–14.Google Scholar
32. Swan, R. G., On seminormality, J. of Alg. 67 (1980), 210–229.Google Scholar
33. Underwood, D. H., On some uniqueness questions in primary representations of ideals, Kyoto Math. J. 35 (1969), 69–94.Google Scholar
34. Vasconcelos, W. V., Divisor theory in module categories, North Holland Mathematical Studies no. 14 (North-Holland, Amsterdam, 1974).Google Scholar