Hostname: page-component-5cf477f64f-r2nwp Total loading time: 0 Render date: 2025-04-02T13:04:24.553Z Has data issue: false hasContentIssue false
  • English
  • Français

NOETHERIAN SPECTRUM ON RINGS AND MODULES

Published online by Cambridge University Press:  01 August 2011

DAVID E. RUSH
DAVID E. RUSH*
Affiliation:
Department of Mathematics, University of California, Riverside, CA 92521-0135, USA e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is shown that the well-known characterizations of when a commutative ring R has Noetherian spectrum carry over to characterizations of when the set Spec(M) of prime submodules of a finitely generated module M is Noetherian. The symmetric algebra SR(M) of M is used to show that the Noetherian property of Spec(R), and some related properties, pass from the ring R to the finitely generated R-modules.

Keywords

13C9913E1513E99
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 53 , Issue 3 , September 2011 , pp. 707 - 715
Copyright
Copyright © Glasgow Mathematical Journal Trust 2011

References

REFERENCES

1.Atiyah, M. F. and MacDonald, I. G., Introduction to commutative algebra (Addison-Wesley, Reading, MA, 1969).Google Scholar
2.Azizi, A., Radical formula and prime submodules, J. Algebra 307 (2007), 454–460.CrossRefGoogle Scholar
3.Azizi, A., Radical formula and weakly prime submodules, Glasgow Math. J. 51 (2009), 405–412.CrossRefGoogle Scholar
4.Bourbaki, N., Commutative algebra (Addison-Wesley, Reading, MA, 1970).Google Scholar
5.Duraivel, T., Topology on spectrum of modules, J. Ramanujan Math. Soc. 9 (1994), 25–34.Google Scholar
6.Gilmer, R. and Heinzer, W., The Laskerian property, power series rings, and Noetherian spectra, Proc. Amer. Math. Soc. 79 (1980), 13–16.CrossRefGoogle Scholar
7.Heinzer, W. and Ohm, J., Locally Noetherian commutative rings, Trans. Amer. Math. Soc. 158 (1971), 273–284.CrossRefGoogle Scholar
8.Iroz, J. and Rush, D. E., Associated prime ideals in non-Noetherian rings, Can. J. Math. XXXVI (1984), 344–360.CrossRefGoogle Scholar
9.Jenkins, J. and Smith, P. F., On the prime radical of a module over a commutative ring, Commun. Algebra 20 (1992), 3593–3602.CrossRefGoogle Scholar
10.Kaplansky, I., Commutative rings (The University of Chicago Press, Chicago, IL, 1974).Google Scholar
11.Karakas, H. I., On Noetherian modules, META J. Pure Appl. Sci. 5 (1972), 165–168.Google Scholar
12.Leung, K. H. and Man, S. H., On commutative Noetherian rings which satisfy the radical formula, Glasgow Math. J. 39 (1997), 285–293.CrossRefGoogle Scholar
13.Lu, C.-P., Prime submodules of modules, Comment. Math. Univ. St. Pauli 33 (1984), 61–69.Google Scholar
14.Lu, C.-P., M-radicals of submodules in modules, Math. Japan 34 (1989), 211–219.Google Scholar
15.Lu, C.-P., M-radicals of submodules in modules II, Math. Japan 35 (1990), 991–1001.Google Scholar
16.Lu, C.-P., Unions of prime submodules, Houston J. Math. 23 (1997), 203–213.Google Scholar
17.Lu, C.-P., Saturations of submodules, Commun. Algebra 31 (2003), 2655–2673.CrossRefGoogle Scholar
18.Man, S. H., On commutative Noetherian rings which satisfy the generalized radical formula, Commun. Algebra 27 (1999), 4075–4088.CrossRefGoogle Scholar
19.Man, S. H. and Smith, P., On chains of prime submodules, Isr. J. Math. 127 (2002), 131–155.CrossRefGoogle Scholar
20.McCasland, R. and Moore, M., On radicals of submodules, Commun. Algebra 19 (1991), 1327–1341.CrossRefGoogle Scholar
21.McCasland, R. and Moore, M., Prime submodules, Commun. Algebra 20 (1992), 1803–1817.CrossRefGoogle Scholar
22.McCasland, R. and Smith, P., On the prime submodules of Noetherian modules, Rocky Mountain J. Math. 23 (1993), 1041–1062.CrossRefGoogle Scholar
23.McCasland, R., Moore, M. and Smith, P., On the spectrum of a module over a commutative ring, Commun. Algebra 25 (1997), 79–103.CrossRefGoogle Scholar
24.McCasland, R. L. and Smith, P. F., Generalised associated primes and radicals of submodules, Int. Electron. J. Algebra 4 (2008), 159–176.Google Scholar
25.Marcelo, A. and Masqué, J. M., Prime submodules, the descent invariant, and modules of finite length, J. Algebra 189 (1997), 273–293.CrossRefGoogle Scholar
26.Marcelo, A. and Rodriguez, C., Radicals of submodules and symmetric algebra, Commun. Algebra 28 (2000), 4611–4617.CrossRefGoogle Scholar
27.Marley, T., On rings for which finitely generated ideals have only finitely many minimal components, Commun. Algebra 35 (2007), 1757–1760.CrossRefGoogle Scholar
28.McCasland, R. and Moore, M., On radicals of submodules of finitely generated modules, Can. Math. Bull. 29 (1986), 37–39.CrossRefGoogle Scholar
29.Ohm, J. and Pendleton, R. L.Rings with Noetherian spectrum, Duke Math. J. 35 (1968), 631–639.CrossRefGoogle Scholar
30.Pusat-Yilmaz, D. and Smith, P., Radicals of submodules of free modules, Comm. Alg. 27 (1999), 2253–2266.CrossRefGoogle Scholar
31.Rush, D. E. and Wallace, L. J., Noetherian maximal spectrum and coprimely packed localizations of polynomial rings, Houston J. Math. 38 (2002), 437–448.Google Scholar
32.Smith, P., Primary modules over Noetherian rings, Glasgow Math. J. 43 (2001), 103–111.CrossRefGoogle Scholar
33.Tiras, Y., Harmanci, A. and Smith, P., A characterization of prime submodules, J. Alg., 212 (1999), 743–752.CrossRefGoogle Scholar
34.Zariski, O. and Samuel, P., Commutative algebra, Vol. 1 (Van Nostrand, Princeton, NJ, 1958).Google Scholar