Research Article
Year 2020, Volume 28, Issue 28, 141 - 155, 14.07.2020

### 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

### 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 Mathematics Articles 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 July 14, 2020 Year 2020, Volume 28, Issue 28