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MODULES WHOSE CLASSICAL PRIME SUBMODULES ARE INTERSECTIONS OF MAXIMAL SUBMODULES
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 Title & Authors
MODULES WHOSE CLASSICAL PRIME SUBMODULES ARE INTERSECTIONS OF MAXIMAL SUBMODULES
Arabi-Kakavand, Marzieh; Behboodi, Mahmood;
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 Abstract
Commutative rings in which every prime ideal is an intersection of maximal ideals are called Hilbert (or Jacobson) rings. We propose to define classical Hilbert modules by the property that classical prime submodules are intersections of maximal submodules. It is shown that all co-semisimple modules as well as all Artinian modules are classical Hilbert modules. Also, every module over a zero-dimensional ring is classical Hilbert. Results illustrating connections amongst the notions of classical Hilbert module and Hilbert ring are also provided. Rings R over which all modules are classical Hilbert are characterized. Furthermore, we determine the Noetherian rings R for which all finitely generated R-modules are classical Hilbert.
 Keywords
Hilbert ring;Hilbert module;classical prime submodule;
 Language
English
 Cited by
 References
1.
S. A. Amitsur and C. Procesi, Jacobson rings and Hilbert algebras with polynomial identities, Ann. Mat. Pura Appl. (4) 71 (1966), 61-72. crossref(new window)

2.
A. Azizi, Weakly prime submodules and prime submodules, Glasg. Math. J. 48 (2006), no. 2, 343-346. crossref(new window)

3.
M. Baziar and M. Behboodi, Classical primary submodules and decomposition theory of modules, J. Algebra Appl. 8 (2009), no. 3, 351-362. crossref(new window)

4.
M. Baziar, M. Behboodi, and H. Sharif, Uniformly classical primary submodules, Comm. Algebra 40 (2012), no. 9, 3192-3201. crossref(new window)

5.
M. Behboodi, Classical prime submodules, Ph.D Thesis, Chamran University Ahvaz Iran 2004.

6.
M. Behboodi, A generalization of the classical krull dimension for modules, J. Algebra 305 (2006), no. 2, 1128-1148. crossref(new window)

7.
M. Behboodi, On weakly prime radical of modules and semi-compatible modules, Acta Math. Hungar. 113 (2006), no. 3, 239-250.

8.
M. Behboodi, A generalization of Baer's lower nilradical for modules, J. Algebra Appl. 6 (2007), no. 2, 337-353. crossref(new window)

9.
M. Behboodi, On the prime radical and Baer's lower nilradical of modules, Acta Math. Hungar. 122 (2009), no. 3, 293-306. crossref(new window)

10.
M. Behboodi and H. Koohy, Weakly prime modules, Vietnam J. Math. 32 (2004), no. 2, 185-195.

11.
M. Behboodi and M. J. Noori, Zariski-like topology on the classical prime spectrum of a module, Bull. Iranian Math. Soc. 35 (2009), no. 1, 255-271.

12.
M. Behboodi and S. H. Shojaei, On chains of classical prime submodules and dimension theory of modules, Bull. Iranian Math. Soc. 36 (2010), no. 1, 149-166.

13.
J. Dauns, Prime modules, J. Reine Angew. Math. 298 (1976), 156-181.

14.
M. Ferrero and M. M. Parmenter, A note on Jacobson rings and polynomial rings, Proc. Amer. Math. Soc. 105 (1989), no. 2, 281-286. crossref(new window)

15.
K. Fujita and S. Itoh, A note on Noetherian Hilbert rings, HiroshimaMath. J. 10 (1980), no. 1, 153-161.

16.
O. Goldman, Hilbert rings and the Hilbert Nullstellensatz, Math. Z. 54 (1951), 136-140. crossref(new window)

17.
K. R. Goodearl and R. B. Warfild, An Introduction to Noncommutative Noetherian Rings, London Mathematical socity. Student Texts 16, Camberidge University Press, Cambrige 1989.

18.
T. Hungerford, Algebra, Springer-verlag 1997.

19.
A. Kaucikas and R.Wisbauer, Noncommutative Hilbert rings. J. Algebra Appl. 3 (2004), no. 4, 437-443. crossref(new window)

20.
M. Maani Shirazi and H. Sharif, Hilbert modules, Int. J. Pure Appl. Math. 20 (2005), no. 1, 1-7.

21.
C. Procesi, Noncommutative Jacobson rings, Ann. Scuola Norm. Sup. Pisa 21 (1967), 281-290.

22.
L. J., Jr. Ratliff, Hilbert rings and the chain condition for prime ideals, J. Reine Agnew. Math. 283/284 (1976), 154-163.

23.
J. F. Watters, Polynomial extensions of Jacobson rings, J. Algebra 36 (1975), no. 2, 302-308. crossref(new window)

24.
J. F. Watters, The Brown-McCoy radical and Jacobson rings, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys. 24 (1976), no. 2, 91-99.

25.
R. Wisbauer, Foundations of Modules and Ring Theory, Gordon and Breach Reading, 1991.