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

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. $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$

Keywords

• Graded ring
• graded module
• graded Prüfer domain
• graded primary decomposition

References

1. D. D. Anderson, D. F. Anderson and G. W. Chang, Graded-valuation domains, Comm. Algebra, 45 (2017), 4018-4029.
2. D. F. Anderson, G. W. Chang and M. Zafrullah, Graded Prüfer domains, Comm. Algebra, 46 (2018), 792-809.
3. 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.
4. N. Bourbaki, Commutative Algebra, Chapters 1-7, Translated from the French, Reprint of the 1972 edition, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989.
5. D. A. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom., 4 (1995), 17-50.
6. P. Dutton, Prime ideals attached to a module, Quart. J. Math. Oxford Ser. (2), 29(116) (1978), 403-413.
7. N. Epstein and J. Shapiro, Strong Krull primes and at modules, J. Pure Appl. Algebra, 218 (2014), 1712-1729.
8. L. Fuchs and E. Mosteig, Ideal theory in Prüfer domains - an unconventional approach, J. Algebra, 252 (2002), 411-430.

9. J. Iroz and D. E. Rush, Associated prime ideals in non-Noetherian rings, Canad. J. Math., 36(2) (1984), 344-360.
10. 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.
11. 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.
12. M. D. Larsen and P. J. McCarthy, Multiplicative Theory of Ideals, Pure and Applied Mathematics, 43, Academic Press, New York-London, 1971.
13. C. Nastasescu and F. Van Oystaeyen, Graded Ring Theory, North-Holland Mathematical Library, 28, North-Holland Publishing Co., Amsterdam-New York, 1982.
14. M. Perling and S. D. Kumar, Primary decomposition over rings graded by finitely generated abelian groups, J. Algebra, 318 (2007), 553-561.
15. M. Perling and G. Trautmann, Equivariant primary decomposition and toric sheaves, Manuscripta Math., 132 (2010), 103-143.
16. M. Refai and K. Al-Zoubi, On graded primary ideals, Turkish J. Math., 28 (2004), 217-229.