GRADED I-PRIME SUBMODULES | ||
| Journal of Algebraic Systems | ||
| دوره 10، شماره 2، فروردین 2023، صفحه 225-243 اصل مقاله (199.06 K) | ||
| نوع مقاله: Original Manuscript | ||
| شناسه دیجیتال (DOI): 10.22044/jas.2022.11158.1556 | ||
| نویسندگان | ||
| I. Akray* 1؛ Sh. Othman2؛ A. Jabbar3؛ H. Hussein1 | ||
| 1Department of Mathematics, Soran University, Erbil, Iraq. | ||
| 2Department of Mathematics, Salahaddin university, Erbil, Iraq. | ||
| 3Department of Mathematics, University of Sulaimani, Erbil, Iraq. | ||
| چکیده | ||
| Let $R= \bigoplus_{g \in G} R_g$ 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 $F \otimes P$ is graded $I-$prime submodule of $F \otimes M$. 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. | ||
| کلیدواژهها | ||
| I-prime ideals؛ I-prime submodule؛ graded prime ideal؛ graded prime submodule | ||
| مراجع | ||
|
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. | ||
|
آمار تعداد مشاهده مقاله: 695 تعداد دریافت فایل اصل مقاله: 432 |
||