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Commutative rings in which every prime ideal is an inter- section 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 clas- sical 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. Further- more, we determine the Noetherian rings R for which all finitely generated R-modules are classical Hilbert.
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참고문헌 (25)
참고문헌 신청
A. Azizi / 2006 / Weakly prime submodules and prime submodules / Glasg. Math. J 48 (2) : 343 ~ 346
A. Kauˇcikas / 2004 / Noncommutative Hilbert rings / J. Algebra Appl 3 (4) : 437 ~ 443
C. Procesi / 1967 / Noncommutative Jacobson rings / Ann. Scuola Norm. Sup. Pisa 21 : 281 ~ 290
J. Dauns / 1976 / Prime modules / J. Reine Angew. Math 298 : 156 ~ 181
J. F. Watters / 1975 / Polynomial extensions of Jacobson rings / J. Algebra 36 (2) : 302 ~ 308
A. Kauˇcikas / 2004 / Noncommutative Hilbert rings / J. Algebra Appl 3 (4) : 437 ~ 443
C. Procesi / 1967 / Noncommutative Jacobson rings / Ann. Scuola Norm. Sup. Pisa 21 : 281 ~ 290
J. Dauns / 1976 / Prime modules / J. Reine Angew. Math 298 : 156 ~ 181
J. F. Watters / 1975 / Polynomial extensions of Jacobson rings / J. Algebra 36 (2) : 302 ~ 308
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