Graded \(I\)-second submodules. (English) Zbl 1481.13003
In [J. Algebra Relat. Top. 4, No. 2, 41–47 (2016; Zbl 1355.13003)], the reviewer defines \(I\)-prime ideals and \(I\)-prime submodules. Then, F. Farshadifar and H. Ansari-Toroghy [Mat. Vesn. 72, No. 1, 58–65 (2020; Zbl 1462.13007)] used this idea for defining \(I\)-second submodules. In this paper, the authors use the same idea and defined the same notion in graded modules and study some properties.
Reviewer: Ismael Akray (Soran)
References:
| [1] | M. Refai, M. Hailat and S. Obiedat, Graded radicals and graded prime spectra, Far East J. Math. Sci. 1 (2000), 59-73. · Zbl 1095.16505 |
| [2] | S. E. Atani, On graded prime submodules, Chiang Mai J. Sci. 33 (2006), no. 1, 3-7. · Zbl 1099.13001 |
| [3] | R. Abu-Dawwas and K. Al-Zoubi, On graded weakly classical prime submodules, Iran. J. Math. Sci. Inform. 12 (2017), no. 1, 153-161. · Zbl 1374.13002 |
| [4] | R. Abu-Dawwas, K. Al-Zoubi and M. Bataineh, Prime submodules of graded modules, Proyecciones 31 (2012), no. 4, 355-361. · Zbl 1408.13001 |
| [5] | K. Al-Zoubi and R. Abu-Dawwas, On graded quasi-prime submodules, Kyungpook Math. J. 55 (2015), 259-266. · Zbl 1325.13006 |
| [6] | K. Al-Zoubi, M. Jaradat and R. Abu-Dawwas, On graded classical prime and graded prime submodules, Bull. Iranian Math. Soc. 41 (2015), no. 1, 205-2013. · Zbl 1335.13002 |
| [7] | S. E. Atani, On graded weakly prime ideals, Turkish J. Math. 30 (2006), 351-358. · Zbl 1109.13001 |
| [8] | S. E. Atani, On graded weakly prime submodules, Int. Math. Forum 1 (2006), no. 2, 61-66. · Zbl 1145.13303 |
| [9] | C. Nastasescu and F. van Oystaeyen, Methods of Graded Rings, Lecture Notes in Mathematics, 1836, Springer-Verlag, Berlin, 2004. · Zbl 1043.16017 |
| [10] | H. Ansari-Toroghy and F. Farshadifar, On graded second modules, Tamkang J. Math. 43 (2012), no. 4, 499-505. · Zbl 1266.13001 |
| [11] | S. Çeken and M. Alkan, On graded second and coprimary modules and graded second representations, Bull. Malaysian Math. Sci. Soc. Ser. 2 38 (2015), no. 4, 1317-1330. · Zbl 1338.16046 |
| [12] | S. E. Atani and F. Farzalipour, On graded secondary modules, Turkish J. Math. 31 (2007), 371-378. · Zbl 1132.13001 |
| [13] | F. Farshadifar and H. Ansari-Toroghy, I-second submodules of a module, Matematicki Vesnik 72 (2020), no. 1, 58-65. · Zbl 1461.13007 |
| [14] | I. Akray, I-prime ideals, J. Algebra Relat. Topics 4 (2016), no. 2, 41-47. · Zbl 1355.13003 |
| [15] | F. Farzalipour and P. Ghiasvand, On the union of graded prime submodules, Thai J. Math. 9 (2011), no. 1, 49-55. · Zbl 1260.13001 |
| [16] | D. Northcott, Lessons on Rings, Modules, and Multiplicities, Cambridge University Press, Cambridge, 1968. · Zbl 0159.33001 |
| [17] | J. Chen and Y. Kim, Graded irreducible modules are irreducible, Comm. Algebra 45 (2017), no. 5, 1907-1913. · Zbl 1386.13001 |
| [18] | H. Ansari-Toroghy and F. Farshadifar, Graded comultiplication modules, Chiang Mai J. Sci. 38 (2011), no. 3, 311-320. · Zbl 1231.16035 |
| [19] | R. Abu-Dawwas and M. Ali, Comultiplication modules over strongly graded rings, Int. J. Pure Appl. Math. 81 (2012), no. 5, 693-699. |
| [20] | R. Abu-Dawwas, M. Bataineh and A. Dakeek, Graded weak comultiplication modules, Hokkaido Math. J. 48 (2019), 253-261. · Zbl 1418.13001 |
| [21] | M. Refai and K. Al-Zoubi, On graded primary ideals, Turkish J. Math. 28 (2004), no. 3, 217-229. · Zbl 1077.13001 |
| [22] | K. H. Oral, Ü. Tekir and A. G. Agargün, On graded prime and primary submodules, Turkish J. Math. 35 (2011), 159-167. · Zbl 1279.13004 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
