GRADED I-PRIME SUBMODULES

Document Type : Original Manuscript

Authors

1 Department of Mathematics, Soran University, Erbil, Iraq.

2 Department of Mathematics, Salahaddin university, Erbil, Iraq.

3 Department of Mathematics, University of Sulaimani, Erbil, Iraq.

Abstract

Let R=⨁g∈GRg be a G−graded commutative ring with identity, I be a graded ideal and let M a G−graded unitary R-module, where G is a semigroup with identity e. We introduce graded I−prime ideals (submodules) as a generalizations of the classical notions of prime ideals (submodules). We show that the new notions inherite the basic properties of the classical ones. In particular, we investigate the localization theory of these two concepts. We prove that for a faithfull flat module F, a graded submodule P of M is I−prime if and only if F⊗P is graded I−prime submodule of F⊗M. As an application, for finitely generated graded module M over Noetherian graded ring R, the completion of graded I−prime submodules is I−prime submodule.

Keywords

References

1. J. Abaffy, C. G. Broyden and E. Spedicato, A class of direct methods for linear equations, Numer. Math., 45 (1984), 361–376.
2. J. Abaffy and E. Spedicato, ABS Projection Algorithms: Mathematical Techniques for Linear and Nonlinear Equations, Ellis Horwood, Chichester, 1989.
3. I. Akray, I-prime ideals, J. Algebra Relat. Topics, 4(2) (2016), 41–47.
4. I. Akray and H. S. Hussein, I-primary submodules, Acad. J. Garmian Univ., 1(11) (2017), 592–599
5. I. Akray and H. S. Hussein, I-prime submodules, Acta Math. Acad. Paedag. Nyiregyhaziensis, 33(2) (2017), 165–173.
6. D. Anderson and E. V. Smith, Weakly prime ideals, Houston J. Math., 29(4) (2003), 831–840.
7. S. E. Atani and F. Farzalipour, On weakly primary ideals, Georgian Math. J., 12(3) (2005), 423–429.
8. S. E. Atani, On graded weakly prime submodules, Int. Math. Forum, 2 (2006), 61–66.
9. S. E. Atani, On graded weakly prime ideals, Turkish J. Math., 30(4) (2006), 351–358.
10. A. Barnard, Multiplication modules, J. Algebra, 71(1) (1981), 174–178.
11. M. Baziar and M. Behboodi, Classical primary submodules and decomposition theory of modules, J. Algebra Appl., 8(03) (2009), 351–362.
12. M. Behboodi and H. Koohy, Weakly prime modules, Vietnam J. Math., 32(2) (2004), 185–195.
13. C. D. Everett, Group-Graded rings and modules, Math. Z., 174 (1980), 241– 262.
14. J. Jenkin and P. F. Smith, On the prime radical of a module over a commutative ring, Comm. Algebra, 20(12) (1992), 3593–3602.
15. C. Lu, Prime submodules of modules, Comment. Math. Univ. St. Pauli, 33(1) (1984), 61–69.
16. Refai M. and Abu-Dawwas R., On generalizations of graded second submodules, Proyecciones, 39(6) (2020), 1537–1554.
17. M. Refai, M. Hailat and S. Obiedat, Graded radicals and graded prime spectra, Far East J. Math. Sci., part 1 (2000), 59–73.
Journal of Algebraic Systems
Volume 10, Issue 2
January 2023
Pages 225-243

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