Let $ R $ be a $ G $ graded commutative ring and $ M $ be a $ G $-graded $ R $-module. The set of all graded second submodules of $ M $ is denoted by $ Spec_G^s(M), $ and it is called the graded second spectrum of $ M $. We discuss graded rings with Noetherian graded prime spectrum. In addition, we introduce the notion of the graded Zariski socle of graded submodules and explore their properties. We also investigate $ Spec^s_G(M) $ with the Zariski topology from the viewpoint of being a Noetherian space.
Citation: Saif Salam, Khaldoun Al-Zoubi. Graded modules with Noetherian graded second spectrum[J]. AIMS Mathematics, 2023, 8(3): 6626-6641. doi: 10.3934/math.2023335
Let $ R $ be a $ G $ graded commutative ring and $ M $ be a $ G $-graded $ R $-module. The set of all graded second submodules of $ M $ is denoted by $ Spec_G^s(M), $ and it is called the graded second spectrum of $ M $. We discuss graded rings with Noetherian graded prime spectrum. In addition, we introduce the notion of the graded Zariski socle of graded submodules and explore their properties. We also investigate $ Spec^s_G(M) $ with the Zariski topology from the viewpoint of being a Noetherian space.
| [1] | R. Abu-Dawwas, M. Ali, Comultiplication modules over strongly graded rings, International Journal of Pure and Applied Mathematics, 81 (2012), 693–699. |
| [2] |
A. J. AL-Juburie, An additional properties of the graded prime spectrum of (R, G), J. Phys.: Conf. Ser., 1591 (2020), 012104. http://doi.org/10.1088/1742-6596/1591/1/012104 doi: 10.1088/1742-6596/1591/1/012104
|
| [3] |
K. Al-Zoubi, A. Al-Qderat, Some properties of graded comultiplication module, Open Math., 15 (2017), 187–192. https://doi.org/10.1515/math-2017-0016 doi: 10.1515/math-2017-0016
|
| [4] | H. Ansari-Toroghy, F. Farshadifar, Graded comultiplication modules, Chiang Mai J. Sci., 38 (2011), 311–320. |
| [5] | H. Ansari-Toroghy, F. Farshadifar, On graded second modules, Tamkang J. Math., 43 (2012), 499–505. https://doi.org/10.5556/j.tkjm.43.2012.1319 |
| [6] | M. F. Atiyah, I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., 1969. |
| [7] | N. Bourbaki, Algebre commutative: Chap 1–2, Paris: Hermann, 1961. |
| [8] |
S. Ceken, M. Alkan, On graded second and coprimary modules and graded secondary representations, Bull. Malays. Math. Sci. Soc., 38 (2015), 1317–1330. http://doi.org/10.1007/s40840-014-0097-6 doi: 10.1007/s40840-014-0097-6
|
| [9] | C. Năstăsescu, F. Van Oystaeyen, Methods of graded rings, Berlin: Springer, 2004. http://doi.org/10.1007/b94904 |
| [10] |
K. H. Oral, Ü. Tekir, A. G. Ağargün, On graded prime and primary submodules, Turkish J. Math., 35 (2011), 159–167. http://doi.org/10.3906/mat-0904-11 doi: 10.3906/mat-0904-11
|
| [11] | N. A. Ozkiırisci, K. H. Oral, Ü. Tekir, Graded prime spectrum of a graded module, Iranian Journal of Science & Technology, 37 (2013), 411–420. |
| [12] | M. Refai, On properties of $G\text{-spec}(R)$, Scientiae Mathematicae Japonicae Online, 4 (2001), 491–495. |
| [13] | M. Refai, K. Al-Zoubi, On graded primary ideals, Turk. J. Math., 28 (2004), 217–229. |
| [14] | M. Refai, M. Hailat, S. Obiedat, Graded radicals and graded prime spectra, Far East Journal of Mathematical Sciences, Special Volume (2000), 59–73. |
| [15] |
S. Salam, K. Al-Zoubi, The Zariski topology on the graded primary spectrum of a graded module over a graded commutative ring, Appl. Gen. Topol., 23 (2022), 345–361. https://doi.org/10.4995/agt.2022.16332 doi: 10.4995/agt.2022.16332
|
| [16] | S. Salam, K. Al-Zoubi, The Zariski topology on the graded second spectrum of a graded module.Research Square, 2022, preprint. |
Saif Salam, Khaldoun Al-Zoubi. Graded modules with Noetherian graded second spectrum[J]. AIMS Mathematics, 2023, 8(3): 6626-6641. doi: 10.3934/math.2023335
DownLoad: