Research Article
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Year 2019, Volume: 25 Issue: 25, 186 - 198, 08.01.2019
https://doi.org/10.24330/ieja.504147

Abstract

References

  • F. Alarcon, D. D. Anderson and C. Jayaram, Some results on abstract com- mutative ideal theory, Period. Math. Hungar., 30(1) (1995), 1-26.
  • E. A. Al-Khouja, Maximal elements and prime elements in lattice modules, Damascus Univ. J. Basic Sci., 19(2) (2003), 9-21.
  • M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969.
  • S. Ballal and V. Kharat, Zariski topology on lattice modules, Asian-Eur. J. Math., 8(4) (2015), 1550066 (10 pp).
  • M. Behboodi and M. R. Haddadi, Classical Zariski topology of modules and spectral spaces I, Int. Electron. J. Algebra, 4 (2008), 104-130.
  • M. Behboodi and M. R. Haddadi, Classical Zariski topology of modules and spectral spaces II, Int. Electron. J. Algebra, 4 (2008), 131-148.
  • M. Behboodi and M. J. Noori, Zariski-like topology on the classical prime spectrum of a module, Bull. Iranian Math. Soc., 35(1) (2009), 253-269.
  • N. Bourbaki, Algebre Commutative, Chap 1-2, Hermann, Paris, 1961.
  • N. Bourbaki, Elements of Mathematics, General topology, Part 1, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1966.
  • F. Callialp, G. Ulucak and U. Tekir, On the Zariski topology over an L-module M, Turkish J. Math., 41(2) (2017), 326-336.
  • K. R. Goodearl and R. B. War eld, Jr., An Introduction to Noncommuta- tive Noetherian Rings, Second edition, London Math. Soc. Student Texts, 16, Cambridge Univ. Press, Cambridge, 2004.
  • M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc., 142 (1969), 43-60.
  • J. A. Johnson, A-adic completions of Noetherian lattice modules, Fund. Math., 66 (1970), 347-373.
  • C. P. Lu, The Zariski topology on the prime spectrum of a module, Houston J. Math., 25(3) (1999), 417-432.
  • C. S. Manjarekar and A. V. Bingi, Absorbing elements in lattice modules, Int. Electron. J. Algebra, 19 (2016), 58-76.
  • J. R. Munkres, Topology: a First Course, Prentice-Hall, Inc. Eglewood Cli s, New Jersey, 1975.
  • N. Phadatare, S. Ballal and V. Kharat, On the second spectrum of lattice modules, Discuss. Math. Gen. Algebra Appl., 37(1) (2017), 59-74.
  • N. Phadatare and V. Kharat, On the second radical elements of lattice modules, Tbilisi Math. J., 11(4) (2018), 165-173.
  • N. Phadatare, V. Kharat and S. Ballal, On the maximal spectrum of lattice modules, Southeast Asian Bull. Math., Accepted.
  • N. K. Thakare and C. S. Manjarekar, Abstract spectral theory: Multiplicative lattices in which every character is contained in a unique maximal character, Algebra and its applications (New Delhi, 1981), Lecture Notes in Pure and Appl. Math., 91, Dekker, New York, (1984), 265-276.
  • N. K. Thakare, C. S. Manjarekar and S. Maeda, Abstract spectral theory II: minimal characters and minimal spectrums of multiplicative lattices, Acta Sci. Math. (Szeged), 52(1-2) (1988), 53-67.

ON THE CLASSICAL PRIME SPECTRUM OF LATTICE MODULES

Year 2019, Volume: 25 Issue: 25, 186 - 198, 08.01.2019
https://doi.org/10.24330/ieja.504147

Abstract

Let $M$ be a lattice module over a $C$-lattice $L$. A proper element $P$ of $M$ is said to be classical prime if for
$a ,b\in L$ and $X\in M, abX\leq P$ implies that $aX\leq P$ or $bX\leq P$. The set of all classical prime elements of $M$, $Spec^{cp}(M)$ is called as classical prime spectrum. In this article, we introduce and study a topology on $Spec^{cp}(M)$, called as Zariski-like topology of $M$. We investigate this topological space from the point of view of spectral spaces. We show that if $M$ has ascending chain condition on classical prime radical elements, then $Spec^{cp}(M)$ with the Zariski-like topology is a spectral space.
 

References

  • F. Alarcon, D. D. Anderson and C. Jayaram, Some results on abstract com- mutative ideal theory, Period. Math. Hungar., 30(1) (1995), 1-26.
  • E. A. Al-Khouja, Maximal elements and prime elements in lattice modules, Damascus Univ. J. Basic Sci., 19(2) (2003), 9-21.
  • M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969.
  • S. Ballal and V. Kharat, Zariski topology on lattice modules, Asian-Eur. J. Math., 8(4) (2015), 1550066 (10 pp).
  • M. Behboodi and M. R. Haddadi, Classical Zariski topology of modules and spectral spaces I, Int. Electron. J. Algebra, 4 (2008), 104-130.
  • M. Behboodi and M. R. Haddadi, Classical Zariski topology of modules and spectral spaces II, Int. Electron. J. Algebra, 4 (2008), 131-148.
  • M. Behboodi and M. J. Noori, Zariski-like topology on the classical prime spectrum of a module, Bull. Iranian Math. Soc., 35(1) (2009), 253-269.
  • N. Bourbaki, Algebre Commutative, Chap 1-2, Hermann, Paris, 1961.
  • N. Bourbaki, Elements of Mathematics, General topology, Part 1, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1966.
  • F. Callialp, G. Ulucak and U. Tekir, On the Zariski topology over an L-module M, Turkish J. Math., 41(2) (2017), 326-336.
  • K. R. Goodearl and R. B. War eld, Jr., An Introduction to Noncommuta- tive Noetherian Rings, Second edition, London Math. Soc. Student Texts, 16, Cambridge Univ. Press, Cambridge, 2004.
  • M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc., 142 (1969), 43-60.
  • J. A. Johnson, A-adic completions of Noetherian lattice modules, Fund. Math., 66 (1970), 347-373.
  • C. P. Lu, The Zariski topology on the prime spectrum of a module, Houston J. Math., 25(3) (1999), 417-432.
  • C. S. Manjarekar and A. V. Bingi, Absorbing elements in lattice modules, Int. Electron. J. Algebra, 19 (2016), 58-76.
  • J. R. Munkres, Topology: a First Course, Prentice-Hall, Inc. Eglewood Cli s, New Jersey, 1975.
  • N. Phadatare, S. Ballal and V. Kharat, On the second spectrum of lattice modules, Discuss. Math. Gen. Algebra Appl., 37(1) (2017), 59-74.
  • N. Phadatare and V. Kharat, On the second radical elements of lattice modules, Tbilisi Math. J., 11(4) (2018), 165-173.
  • N. Phadatare, V. Kharat and S. Ballal, On the maximal spectrum of lattice modules, Southeast Asian Bull. Math., Accepted.
  • N. K. Thakare and C. S. Manjarekar, Abstract spectral theory: Multiplicative lattices in which every character is contained in a unique maximal character, Algebra and its applications (New Delhi, 1981), Lecture Notes in Pure and Appl. Math., 91, Dekker, New York, (1984), 265-276.
  • N. K. Thakare, C. S. Manjarekar and S. Maeda, Abstract spectral theory II: minimal characters and minimal spectrums of multiplicative lattices, Acta Sci. Math. (Szeged), 52(1-2) (1988), 53-67.
There are 21 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Pradip Girase This is me

Vandeo Borkar

Narayan Phadatare This is me

Publication Date January 8, 2019
Published in Issue Year 2019 Volume: 25 Issue: 25

Cite

APA Girase, P., Borkar, V., & Phadatare, N. (2019). ON THE CLASSICAL PRIME SPECTRUM OF LATTICE MODULES. International Electronic Journal of Algebra, 25(25), 186-198. https://doi.org/10.24330/ieja.504147
AMA Girase P, Borkar V, Phadatare N. ON THE CLASSICAL PRIME SPECTRUM OF LATTICE MODULES. IEJA. January 2019;25(25):186-198. doi:10.24330/ieja.504147
Chicago Girase, Pradip, Vandeo Borkar, and Narayan Phadatare. “ON THE CLASSICAL PRIME SPECTRUM OF LATTICE MODULES”. International Electronic Journal of Algebra 25, no. 25 (January 2019): 186-98. https://doi.org/10.24330/ieja.504147.
EndNote Girase P, Borkar V, Phadatare N (January 1, 2019) ON THE CLASSICAL PRIME SPECTRUM OF LATTICE MODULES. International Electronic Journal of Algebra 25 25 186–198.
IEEE P. Girase, V. Borkar, and N. Phadatare, “ON THE CLASSICAL PRIME SPECTRUM OF LATTICE MODULES”, IEJA, vol. 25, no. 25, pp. 186–198, 2019, doi: 10.24330/ieja.504147.
ISNAD Girase, Pradip et al. “ON THE CLASSICAL PRIME SPECTRUM OF LATTICE MODULES”. International Electronic Journal of Algebra 25/25 (January 2019), 186-198. https://doi.org/10.24330/ieja.504147.
JAMA Girase P, Borkar V, Phadatare N. ON THE CLASSICAL PRIME SPECTRUM OF LATTICE MODULES. IEJA. 2019;25:186–198.
MLA Girase, Pradip et al. “ON THE CLASSICAL PRIME SPECTRUM OF LATTICE MODULES”. International Electronic Journal of Algebra, vol. 25, no. 25, 2019, pp. 186-98, doi:10.24330/ieja.504147.
Vancouver Girase P, Borkar V, Phadatare N. ON THE CLASSICAL PRIME SPECTRUM OF LATTICE MODULES. IEJA. 2019;25(25):186-98.

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