Let $ R $ be a commutative ring with identity and $ N $ be a submodule of an $ R $-module $ M $. We say a nonnil submodule $ N $ of an $ R $-module $ M $ is a $ \mathrm{\Phi} $-powerful (resp., $ \mathrm{\Phi} $-strongly prime) submodule, if $ \mathrm{\Phi}(N) $ is a powerful (resp., strongly prime) submodule of a module $ \mathrm{\Phi}(M) $. We show that a nonnil prime submodule $ N $ of an $ R $-module $ M $ is a $ \mathrm{\Phi} $-powerful submodule if and only if it is a $ \mathrm{\Phi} $-strongly prime submodule. Similarly, if every prime submodule of an $ R $-module $ M $ is a $ \mathrm{\Phi} $-strongly prime, then we call it a $ \mathrm{\Phi} $-pseudo-valuation module ($ \mathrm{\Phi} $-PVM). We also prove that a faithful multiplication $ R $-module $ M $ is $ \mathrm{\Phi} $-PVM if and only if some maximal nonnil submodules of $ M $ are $ \mathrm{\Phi} $-powerful. In this perspective, we analyze that $ M $ is $ \mathrm{\Phi} $-PVM if and only if $ R $ is a PVD. In due course, we provide some characterizations of these submodules along with their relationships under certain conditions.
Citation: Waheed Ahmad Khan, Kiran Farid, Abdelghani Taouti. On $ \Phi $-powerful submodules and $ \mathrm{\Phi} $-strongly prime submodules[J]. AIMS Mathematics, 2021, 6(10): 11610-11619. doi: 10.3934/math.2021674
Let $ R $ be a commutative ring with identity and $ N $ be a submodule of an $ R $-module $ M $. We say a nonnil submodule $ N $ of an $ R $-module $ M $ is a $ \mathrm{\Phi} $-powerful (resp., $ \mathrm{\Phi} $-strongly prime) submodule, if $ \mathrm{\Phi}(N) $ is a powerful (resp., strongly prime) submodule of a module $ \mathrm{\Phi}(M) $. We show that a nonnil prime submodule $ N $ of an $ R $-module $ M $ is a $ \mathrm{\Phi} $-powerful submodule if and only if it is a $ \mathrm{\Phi} $-strongly prime submodule. Similarly, if every prime submodule of an $ R $-module $ M $ is a $ \mathrm{\Phi} $-strongly prime, then we call it a $ \mathrm{\Phi} $-pseudo-valuation module ($ \mathrm{\Phi} $-PVM). We also prove that a faithful multiplication $ R $-module $ M $ is $ \mathrm{\Phi} $-PVM if and only if some maximal nonnil submodules of $ M $ are $ \mathrm{\Phi} $-powerful. In this perspective, we analyze that $ M $ is $ \mathrm{\Phi} $-PVM if and only if $ R $ is a PVD. In due course, we provide some characterizations of these submodules along with their relationships under certain conditions.
| [1] |
M. M. Ali, Idempotent and nilpotent submodules of multiplication modules, Commun. Algebra, 36 (2008), 4620–4642. doi: 10.1080/00927870802186805
|
| [2] | D. F. Anderson, A. Badawi, On $\phi$-Prüfer rings and $\phi$-Bezout rings, Houston J. Math., 30 (2004), 331–343. |
| [3] | D. F. Anderson, A. Badawi, On $\phi$-Dedekind rings and $\phi$-Krull rings, Houston J. Math., 31 (2005), 1007–1022. |
| [4] |
S. E. Atani, S. D. Pishhesari, M. Khoramdel, Some remarks on Prüfer modules, Discuss. Math. Gen. Algebra Appl., 33 (2013), 121–128. doi: 10.7151/dmgaa.1201
|
| [5] | A. Badawi, On $\phi$-Pseudo-valuation rings, In: Advances in commutative ring theory, New York/Basel: Dekker, 1999,101–110. |
| [6] | A. Badawi, On $\phi$-Mori rings, Houston J. Math., 32 (2006), 1–32. |
| [7] | A. Y. Darani, M. Rahmatinia, On $\mathrm{\Phi}$-Mori modules, New York J. Math., 21 (2015), 1269–1282. |
| [8] | A. Y. Darani, Nonnil-noetherian modules over commutative rings, J. Algebr. Syst., 3 (2016), 201–210. |
| [9] | A. Y. Darani, The study of $\phi$-Powerful ideals, Research project at Faculty of Science, Department of Mathematics and Applications, University of Mohaghegh Ardabili, 2019. Available from: http://repository.uma.ac.ir/id/eprint/8260. |
| [10] |
N. J. Groenewald, D. Ssevviiri, Generalization of nilpotency of ring elements to module elements, Commun. Algebra, 42 (2014), 571–577. doi: 10.1080/00927872.2012.718822
|
| [11] | J. R. Hedstrom, E. G. Houston, Pseudo-valuation domains, Pac. J. Math., 75 (1978), 137–147. |
| [12] | A. Khaksari, S. Mehry, R. Safakish, On special submodule of modules, B. Iran. Math. Soc., 40 (2014), 1441–1451. |
| [13] | R. Kumar, A. Gaur, A note on pairs of rings with same prime ideals, 2020, arXiv: 2005.05959v1. |
| [14] | J. Moghaderi, R. Nekooei, Strongly prime submodules and pseudo-valuation modules, Int. Electron. J. Algebra, 10 (2011), 65–75. |
| [15] |
H. Mostafanasab, A. Y. Darani, On $\phi$-n-absorbing primary ideals of commutative rings, J. Korean Math. Soc., 53 (2016), 549–582. doi: 10.4134/JKMS.j150171
|
| [16] |
S. Motmaen, A. Y. Darani, On $\mathrm{\Phi}$-Dedekind, $\mathrm{\Phi}$-Prüfer and $\mathrm{\Phi}$-Bezout modules, Georgian Math. J., 27 (2020), 103–110. doi: 10.1515/gmj-2017-0049
|
| [17] | A. G. Naoum, F. H. Al-Alwan, Dedekind modules, Commun. Algebra, 24 (1996), 397–412. |
Waheed Ahmad Khan, Kiran Farid, Abdelghani Taouti. On $ \Phi $-powerful submodules and $ \mathrm{\Phi} $-strongly prime submodules[J]. AIMS Mathematics, 2021, 6(10): 11610-11619. doi: 10.3934/math.2021674
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