Research Article
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Year 2020, Volume 28, Issue 28, 141 - 155, 14.07.2020
https://doi.org/10.24330/ieja.768206

Abstract

References

  • D. D. Anderson, D. F. Anderson and G. W. Chang, Graded-valuation domains, Comm. Algebra, 45 (2017), 4018-4029.
  • D. F. Anderson, G. W. Chang and M. Zafrullah, Graded Prüfer domains, Comm. Algebra, 46 (2018), 792-809.
  • S. Behara and S. D. Kumar, Group graded associated ideals with at base change of rings and short exact sequences, Proc. Indian Acad. Sci. Math. Sci., 121 (2011), 111-120.
  • N. Bourbaki, Commutative Algebra, Chapters 1-7, Translated from the French, Reprint of the 1972 edition, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989.
  • D. A. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom., 4 (1995), 17-50.
  • P. Dutton, Prime ideals attached to a module, Quart. J. Math. Oxford Ser. (2), 29(116) (1978), 403-413.
  • N. Epstein and J. Shapiro, Strong Krull primes and at modules, J. Pure Appl. Algebra, 218 (2014), 1712-1729.
  • L. Fuchs and E. Mosteig, Ideal theory in Prüfer domains - an unconventional approach, J. Algebra, 252 (2002), 411-430.
  • J. Iroz and D. E. Rush, Associated prime ideals in non-Noetherian rings, Canad. J. Math., 36(2) (1984), 344-360.
  • H. A. Khashan, Graded rings in which every graded ideal is a product of Gr-primary ideals, Int. J. Algebra, 2(13-16) (2008), 779-788.
  • S. D. Kumar and S. Behara, Uniqueness of graded primary decomposition of modules graded over finitely generated abelian groups, Comm. Algebra, 39(7) (2011), 2607-2614.
  • M. D. Larsen and P. J. McCarthy, Multiplicative Theory of Ideals, Pure and Applied Mathematics, 43, Academic Press, New York-London, 1971.
  • C. Nastasescu and F. Van Oystaeyen, Graded Ring Theory, North-Holland Mathematical Library, 28, North-Holland Publishing Co., Amsterdam-New York, 1982.
  • M. Perling and S. D. Kumar, Primary decomposition over rings graded by finitely generated abelian groups, J. Algebra, 318 (2007), 553-561.
  • M. Perling and G. Trautmann, Equivariant primary decomposition and toric sheaves, Manuscripta Math., 132 (2010), 103-143.
  • M. Refai and K. Al-Zoubi, On graded primary ideals, Turkish J. Math., 28 (2004), 217-229.

DIFFERENT TYPES OF G-PRIME IDEALS ASSOCIATED TO A GRADED MODULE AND GRADED PRIMARY DECOMPOSITION IN A GRADED PRÜFER DOMAIN

Year 2020, Volume 28, Issue 28, 141 - 155, 14.07.2020
https://doi.org/10.24330/ieja.768206

Abstract

In this paper, we introduce the notion of graded Prüfer domain as a generalization of Prüfer domain to the graded case. We generalize several types of prime ideals associated to a module over a ring to the graded case and prove that most of them coincide over a graded Prüfer domain. Moreover, we investigate the graded primary decomposition of graded ideals in a graded Prüfer domain under certain conditions and give some applications of it. $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$

References

  • D. D. Anderson, D. F. Anderson and G. W. Chang, Graded-valuation domains, Comm. Algebra, 45 (2017), 4018-4029.
  • D. F. Anderson, G. W. Chang and M. Zafrullah, Graded Prüfer domains, Comm. Algebra, 46 (2018), 792-809.
  • S. Behara and S. D. Kumar, Group graded associated ideals with at base change of rings and short exact sequences, Proc. Indian Acad. Sci. Math. Sci., 121 (2011), 111-120.
  • N. Bourbaki, Commutative Algebra, Chapters 1-7, Translated from the French, Reprint of the 1972 edition, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989.
  • D. A. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom., 4 (1995), 17-50.
  • P. Dutton, Prime ideals attached to a module, Quart. J. Math. Oxford Ser. (2), 29(116) (1978), 403-413.
  • N. Epstein and J. Shapiro, Strong Krull primes and at modules, J. Pure Appl. Algebra, 218 (2014), 1712-1729.
  • L. Fuchs and E. Mosteig, Ideal theory in Prüfer domains - an unconventional approach, J. Algebra, 252 (2002), 411-430.
  • J. Iroz and D. E. Rush, Associated prime ideals in non-Noetherian rings, Canad. J. Math., 36(2) (1984), 344-360.
  • H. A. Khashan, Graded rings in which every graded ideal is a product of Gr-primary ideals, Int. J. Algebra, 2(13-16) (2008), 779-788.
  • S. D. Kumar and S. Behara, Uniqueness of graded primary decomposition of modules graded over finitely generated abelian groups, Comm. Algebra, 39(7) (2011), 2607-2614.
  • M. D. Larsen and P. J. McCarthy, Multiplicative Theory of Ideals, Pure and Applied Mathematics, 43, Academic Press, New York-London, 1971.
  • C. Nastasescu and F. Van Oystaeyen, Graded Ring Theory, North-Holland Mathematical Library, 28, North-Holland Publishing Co., Amsterdam-New York, 1982.
  • M. Perling and S. D. Kumar, Primary decomposition over rings graded by finitely generated abelian groups, J. Algebra, 318 (2007), 553-561.
  • M. Perling and G. Trautmann, Equivariant primary decomposition and toric sheaves, Manuscripta Math., 132 (2010), 103-143.
  • M. Refai and K. Al-Zoubi, On graded primary ideals, Turkish J. Math., 28 (2004), 217-229.

Details

Primary Language English
Subjects Mathematics
Journal Section Articles
Authors

Ajim Uddin ANSARI This is me (Primary Author)
University of Allahabad
India


B. K. SHARMA This is me
University of Allahabad
India


Shiv Datt KUMAR This is me
Motilal Nehru National Institute of Technology
India

Publication Date July 14, 2020
Published in Issue Year 2020, Volume 28, Issue 28

Cite

Bibtex @research article { ieja768206, journal = {International Electronic Journal of Algebra}, issn = {1306-6048}, eissn = {1306-6048}, address = {1710 Sokak, No:41, Batikent/Ankara}, publisher = {Abdullah HARMANCI}, year = {2020}, volume = {28}, number = {28}, pages = {141 - 155}, doi = {10.24330/ieja.768206}, title = {DIFFERENT TYPES OF G-PRIME IDEALS ASSOCIATED TO A GRADED MODULE AND GRADED PRIMARY DECOMPOSITION IN A GRADED PRÜFER DOMAIN}, key = {cite}, author = {Ansarı, Ajim Uddin and Sharma, B. K. and Kumar, Shiv Datt} }
APA Ansarı, A. U. , Sharma, B. K. & Kumar, S. D. (2020). DIFFERENT TYPES OF G-PRIME IDEALS ASSOCIATED TO A GRADED MODULE AND GRADED PRIMARY DECOMPOSITION IN A GRADED PRÜFER DOMAIN . International Electronic Journal of Algebra , 28 (28) , 141-155 . DOI: 10.24330/ieja.768206
MLA Ansarı, A. U. , Sharma, B. K. , Kumar, S. D. "DIFFERENT TYPES OF G-PRIME IDEALS ASSOCIATED TO A GRADED MODULE AND GRADED PRIMARY DECOMPOSITION IN A GRADED PRÜFER DOMAIN" . International Electronic Journal of Algebra 28 (2020 ): 141-155 <https://dergipark.org.tr/en/pub/ieja/issue/55997/768206>
Chicago Ansarı, A. U. , Sharma, B. K. , Kumar, S. D. "DIFFERENT TYPES OF G-PRIME IDEALS ASSOCIATED TO A GRADED MODULE AND GRADED PRIMARY DECOMPOSITION IN A GRADED PRÜFER DOMAIN". International Electronic Journal of Algebra 28 (2020 ): 141-155
RIS TY - JOUR T1 - DIFFERENT TYPES OF G-PRIME IDEALS ASSOCIATED TO A GRADED MODULE AND GRADED PRIMARY DECOMPOSITION IN A GRADED PRÜFER DOMAIN AU - Ajim UddinAnsarı, B. K.Sharma, Shiv DattKumar Y1 - 2020 PY - 2020 N1 - doi: 10.24330/ieja.768206 DO - 10.24330/ieja.768206 T2 - International Electronic Journal of Algebra JF - Journal JO - JOR SP - 141 EP - 155 VL - 28 IS - 28 SN - 1306-6048-1306-6048 M3 - doi: 10.24330/ieja.768206 UR - https://doi.org/10.24330/ieja.768206 Y2 - 2022 ER -
EndNote %0 International Electronic Journal of Algebra DIFFERENT TYPES OF G-PRIME IDEALS ASSOCIATED TO A GRADED MODULE AND GRADED PRIMARY DECOMPOSITION IN A GRADED PRÜFER DOMAIN %A Ajim Uddin Ansarı , B. K. Sharma , Shiv Datt Kumar %T DIFFERENT TYPES OF G-PRIME IDEALS ASSOCIATED TO A GRADED MODULE AND GRADED PRIMARY DECOMPOSITION IN A GRADED PRÜFER DOMAIN %D 2020 %J International Electronic Journal of Algebra %P 1306-6048-1306-6048 %V 28 %N 28 %R doi: 10.24330/ieja.768206 %U 10.24330/ieja.768206
ISNAD Ansarı, Ajim Uddin , Sharma, B. K. , Kumar, Shiv Datt . "DIFFERENT TYPES OF G-PRIME IDEALS ASSOCIATED TO A GRADED MODULE AND GRADED PRIMARY DECOMPOSITION IN A GRADED PRÜFER DOMAIN". International Electronic Journal of Algebra 28 / 28 (July 2020): 141-155 . https://doi.org/10.24330/ieja.768206
AMA Ansarı A. U. , Sharma B. K. , Kumar S. D. DIFFERENT TYPES OF G-PRIME IDEALS ASSOCIATED TO A GRADED MODULE AND GRADED PRIMARY DECOMPOSITION IN A GRADED PRÜFER DOMAIN. IEJA. 2020; 28(28): 141-155.
Vancouver Ansarı A. U. , Sharma B. K. , Kumar S. D. DIFFERENT TYPES OF G-PRIME IDEALS ASSOCIATED TO A GRADED MODULE AND GRADED PRIMARY DECOMPOSITION IN A GRADED PRÜFER DOMAIN. International Electronic Journal of Algebra. 2020; 28(28): 141-155.
IEEE A. U. Ansarı , B. K. Sharma and S. D. Kumar , "DIFFERENT TYPES OF G-PRIME IDEALS ASSOCIATED TO A GRADED MODULE AND GRADED PRIMARY DECOMPOSITION IN A GRADED PRÜFER DOMAIN", International Electronic Journal of Algebra, vol. 28, no. 28, pp. 141-155, Jul. 2020, doi:10.24330/ieja.768206