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Primitive ideals of nice Ore localizations

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1448)

Abstract

Let R be a ring and let r be an hereditary torsion theory such that R has the descending chain condition on r-closed left ideals. Let {P 1, P 2 ·, P n } be a link closed set of r-closed prime ideals and C be a left reversible left Ore set in \(C\left( {\bigcap\limits_{i = 1}^n {P_i } } \right)\). If C −1 R is left artinian, then r can be modified to another torsion theory σ such that R has DCC on σ-closed left ideals and the primitive ideals of C −1 R are precisely localizations of the σ-closed prime ideals of R. As an application, new information about localization at sets of minimal primes in rings with left Krull dimension and in Noetherian rings is obtained. The condition that the primitive ideals of C −1 R are precisely {C −1 P 1, C −1 P 2, ·, C −1 P n } is also studied.

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References

  1. T. Albu, F-semicocritical modules, F-primitive ideals and prime ideals, Rev. Roumaine Math. Pures Appl. 31 (1986) 449–459.

    MathSciNet  MATH  Google Scholar 

  2. T. Albu and C. NĂstĂsescu, Relative finiteness in module theory, Texts in Pure and Appl. Math. 84, Marcel Dekker, New York, 1984.

    MATH  Google Scholar 

  3. A.K. Boyle and K.A. Kosler, Localization at collections of minimal prime, J. Algebra 119 (1988) 147–161.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. J.S. Golan, Torsion theories, Pitman monographs and surveys in pure and appl. Math., Longman Scientific Publishing, London, 1986.

    Google Scholar 

  5. K.R. Goodearl, Ring theory, Texts in Pure and Appl. Math. 33, Marcel Dekker, New York, 1976.

    MATH  Google Scholar 

  6. A.V. Jategaonkar, Localizations in Noetherian rings, London Math. Soc. Lecture Notes 98, Cambridge University Press, London, 1985.

    Google Scholar 

  7. C. Lanski, Nil subrings of Goldie rings are nilpotent, Can. J. Math. 21 (1969) 904–907.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. N.H. McCoy, The theory of rings, Macmillan, New York, 1964.

    MATH  Google Scholar 

  9. B. Stenström, Rings and modules of quotients, Lecture Notes in Math. 237, Springer-Verlag, Berlin, 1971.

    MATH  Google Scholar 

  10. M.L. Teply, Modules semicocritical with respect to a torsion theory and their applications, Israel J. Math. 54 (1986) 181–200.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. M.L. Teply, Semicocritical modules, University of Murcia Publications, Murcia, Spain, 1987.

    MATH  Google Scholar 

  12. M.L. Teply, Links, Ore sets and classical localizations, preprint, 1989.

    Google Scholar 

  13. B. Torrecillas, Links between closed prime ideals, preprint.

    Google Scholar 

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© 1990 Springer-Verlag

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Teply, M.L., Torrecillas, B. (1990). Primitive ideals of nice Ore localizations. In: Jain, S.K., López-Permouth, S.R. (eds) Non-Commutative Ring Theory. Lecture Notes in Mathematics, vol 1448. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0091251

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  • DOI: https://doi.org/10.1007/BFb0091251

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-53164-7

  • Online ISBN: 978-3-540-46745-8

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