Graded strongly prime submodules over graded commutative rings

Subject Areas : Algebra

Farkhondeh Farzalipour ¹ (Department of Mathematics, Payame Noor University, Tehran, Iran)
Peyman Ghiasvand ² (Department of Mathematics, Payame Noor University, Tehran, Iran)
Masoomeh Hezarjaribi Dastaki ³ (Department of Mathematics, Payame Noor University, Tehran, Iran)

Received: 2022-09-21 Accepted : 2023-01-10 Published : 2024-01-18

Keywords: زیرمدول اول مدرج, زیرمدول قویاً اول مدرج, زیرمدول قویاً نیم‌اول مدرج,

Abstract :

Let G be a group with identity e, R a graded ring and M a graded R-module. ‎A‎ proper graded submodule ‎ N of ‎ M ‎is said ‎‎to ‎be‎ graded strongly ‎‎prime,‎ if we have ((N+Rx_g):_R⁡M)y_h⊆N, then x_g∈ N or y_h∈N ‎‎for all x_g,y_h∈h(M)‎‎‎‎‎. In this paper, we introduce the concept of graded strongly prime submodules as a generalization of graded prime submodules and we investigate some examples and basic properties of graded strongly prime submodules and state new results in this regard. In fact, in this article we show that the concept of graded strongly prime submodules is different from the concept of graded prime submodules. In continuing, we study the behavior of this structure module homomorphis, localization, quotient modules, Cartesian product. Finally, we state two kind of graded submodules of the amalgamation module along a graded ideal and investigate conditions under which they are graded strongly prime.

References:


[1] R. Abu-Dawwas, ‎M. Bataineh. ‎Graded prime submodules over non-commutative rings. ‎Vietnam J‎. ‎Math.‎ ‎46(3): 681-692‎(2018).‎‎

[2] ‎S. Ebrahimi Atani, F. Farzalipour‎. ‎‎On graded secondary modules‎. ‎Turk‎. ‎J‎. ‎Math.‎ ‎31: ‎371-378(2007)‎.

[3] J. ‎Escoriza, B. T‎orrecillas‎‎‎. ‎‎‎Multiplication ‎objects in commutative grothendieck categories‎‎. ‎Comm. Algebra‎‎‎ ‎26‎(‎6‎): ‎1867‎-‎1883(‎1998‎)‎‎‎.

[4] ‎F. Farzalipour, P. Ghiasvand‎. ‎On the union of graded prime submodules‎. ‎Thai‎. ‎J‎. ‎Math.‎ 9(1): ‎49-55(2011)‎‎.

[5] ‎F. Farzalipour, P. Ghiasvand‎, ‎M. Adlifard.‎ On graded weakly semiprime submodules‎‎. Thai‎. ‎J‎. Math.‎ 12(1): ‎167-174(2014)‎‎.

[6] ‎P. Ghiasvand, F. Farzalipour ‎.‎‎ Graded semiprime submodules over non-commutative graded rings‎. ‎J. Algebraic System‎ 10(1):95-110(2022)‎‎.

[7] P. Ghiasvand, F. ‎Farzalipour.‎ ‎‎On graded weak multiplication modules‎‎. ‎Tamkang J‎. ‎Math.‎ ‎43(2): ‎171-177‎(2012)‎.

[8] K. Hakan Oral, U. Tekir, A. G. ‎‎ Ag‎‎‎arg‎u‎‎‎‎‎n‎‎‎‎‎. On graded prime and primary submodules. ‎Turk. J. ‎Math.‎ ‎35:‎ ‎1‎59-167‎‎(2011)‎.

[9] N. Nastasescu, F. ‎Van Oystaeyen, ‎Graded Rings Theory‎. ‎Mathematical Library 28‎, ‎North Holand‎, ‎Amsterdam‎, ‎(19‎82‎)‎.

[10] ‎N Nastasescu, F. ‎Van Oystaeyen‎, ‎Methods of Graded Rings‎. Lecture Notes in Mathematics, ‎vol‎. ‎1836‎. ‎Springer‎, (2004).

[11] M. Refaei, K. ‎Alzobi‎. ‎On graded primary ideals. Turk‎. ‎J‎. ‎Math.‎ 28(3): ‎217-229(2004)‎‎.