Geodesic
NOETHERIAN SPECTRUM ON RINGS AND MODULES
Glasgow mathematical journal, Tome 53 (2011) no. 3, pp. 707-715

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

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.
DOI : 10.1017/S0017089511000280
Mots-clés : 13C99, 13E15, 13E99 
RUSH, DAVID E. NOETHERIAN SPECTRUM ON RINGS AND MODULES. Glasgow mathematical journal, Tome 53 (2011) no. 3, pp. 707-715. doi: 10.1017/S0017089511000280
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[1] 1.Atiyah, M. F. and MacDonald, I. G., Introduction to commutative algebra (Addison-Wesley, Reading, MA, 1969). Google Scholar

[2] 2.Azizi, A., Radical formula and prime submodules, J. Algebra 307 (2007), 454–460. Google Scholar | DOI

[3] 3.Azizi, A., Radical formula and weakly prime submodules, Glasgow Math. J. 51 (2009), 405–412. Google Scholar | DOI

[4] 4.Bourbaki, N., Commutative algebra (Addison-Wesley, Reading, MA, 1970). Google Scholar

[5] 5.Duraivel, T., Topology on spectrum of modules, J. Ramanujan Math. Soc. 9 (1994), 25–34. Google Scholar

[6] 6.Gilmer, R. and Heinzer, W., The Laskerian property, power series rings, and Noetherian spectra, Proc. Amer. Math. Soc. 79 (1980), 13–16. Google Scholar | DOI

[7] 7.Heinzer, W. and Ohm, J., Locally Noetherian commutative rings, Trans. Amer. Math. Soc. 158 (1971), 273–284. Google Scholar | DOI

[8] 8.Iroz, J. and Rush, D. E., Associated prime ideals in non-Noetherian rings, Can. J. Math. XXXVI (1984), 344–360. Google Scholar | DOI

[9] 9.Jenkins, J. and Smith, P. F., On the prime radical of a module over a commutative ring, Commun. Algebra 20 (1992), 3593–3602. Google Scholar | DOI

[10] 10.Kaplansky, I., Commutative rings (The University of Chicago Press, Chicago, IL, 1974). Google Scholar

[11] 11.Karakas, H. I., On Noetherian modules, META J. Pure Appl. Sci. 5 (1972), 165–168. Google Scholar

[12] 12.Leung, K. H. and Man, S. H., On commutative Noetherian rings which satisfy the radical formula, Glasgow Math. J. 39 (1997), 285–293. Google Scholar | DOI

[13] 13.Lu, C.-P., Prime submodules of modules, Comment. Math. Univ. St. Pauli 33 (1984), 61–69. Google Scholar

[14] 14.Lu, C.-P., M-radicals of submodules in modules, Math. Japan 34 (1989), 211–219. Google Scholar

[15] 15.Lu, C.-P., M-radicals of submodules in modules II, Math. Japan 35 (1990), 991–1001. Google Scholar

[16] 16.Lu, C.-P., Unions of prime submodules, Houston J. Math. 23 (1997), 203–213. Google Scholar

[17] 17.Lu, C.-P., Saturations of submodules, Commun. Algebra 31 (2003), 2655–2673. Google Scholar | DOI

[18] 18.Man, S. H., On commutative Noetherian rings which satisfy the generalized radical formula, Commun. Algebra 27 (1999), 4075–4088. Google Scholar | DOI

[19] 19.Man, S. H. and Smith, P., On chains of prime submodules, Isr. J. Math. 127 (2002), 131–155. Google Scholar | DOI

[20] 20.McCasland, R. and Moore, M., On radicals of submodules, Commun. Algebra 19 (1991), 1327–1341. Google Scholar | DOI

[21] 21.McCasland, R. and Moore, M., Prime submodules, Commun. Algebra 20 (1992), 1803–1817. Google Scholar | DOI

[22] 22.McCasland, R. and Smith, P., On the prime submodules of Noetherian modules, Rocky Mountain J. Math. 23 (1993), 1041–1062. Google Scholar | DOI

[23] 23.McCasland, R., Moore, M. and Smith, P., On the spectrum of a module over a commutative ring, Commun. Algebra 25 (1997), 79–103. Google Scholar | DOI

[24] 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] 25.Marcelo, A. and Masqué, J. M., Prime submodules, the descent invariant, and modules of finite length, J. Algebra 189 (1997), 273–293. Google Scholar | DOI

[26] 26.Marcelo, A. and Rodriguez, C., Radicals of submodules and symmetric algebra, Commun. Algebra 28 (2000), 4611–4617. Google Scholar | DOI

[27] 27.Marley, T., On rings for which finitely generated ideals have only finitely many minimal components, Commun. Algebra 35 (2007), 1757–1760. Google Scholar | DOI

[28] 28.McCasland, R. and Moore, M., On radicals of submodules of finitely generated modules, Can. Math. Bull. 29 (1986), 37–39. Google Scholar | DOI

[29] 29.Ohm, J. and Pendleton, R. L.Rings with Noetherian spectrum, Duke Math. J. 35 (1968), 631–639. Google Scholar | DOI

[30] 30.Pusat-Yilmaz, D. and Smith, P., Radicals of submodules of free modules, Comm. Alg. 27 (1999), 2253–2266. Google Scholar | DOI

[31] 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] 32.Smith, P., Primary modules over Noetherian rings, Glasgow Math. J. 43 (2001), 103–111. Google Scholar | DOI

[33] 33.Tiras, Y., Harmanci, A. and Smith, P., A characterization of prime submodules, J. Alg., 212 (1999), 743–752. Google Scholar | DOI

[34] 34.Zariski, O. and Samuel, P., Commutative algebra, Vol. 1 (Van Nostrand, Princeton, NJ, 1958). Google Scholar

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