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On Divided Modules


Recall that a commutative ring R is said to be a divided ring if its each prime ideal P is comparable with each principal ideal (a), where \(a\in R.\) In this paper, we extend the notion of divided rings to modules in two different ways: let R be a commutative ring with identity and M a unital R-module. Then M is said to be a divided (weakly divided) module if its each prime submodule N of M is comparable with each cyclic submodule Rm (rM) of M, where \(m\in M\) (\(r\in R)\). In addition to give many characterizations of divided modules, some topological properties of (quasi-) Zariski topology of divided modules are investigated. Also, we study the divided property of trivial extension \(R\ltimes M\).

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  1. Ali MM (2009) Invertibility of multiplication modules III. N Z J Math 39:193–213

  2. Anderson DD, Winders M (2009) Idealization of a module. J Commut Algebra 1(1):3–56.

  3. Ayache A, Dobbs DE (2016) Strongly divided domains. Ricerche Mat 65(1):127–154.

  4. Ayache A, Dobbs DE (2017) Strongly divided rings with zero-divisors. Palest J Math 6(2):380–395

  5. Ayache A, Dobbs DE (2019) Strongly divided pairs of integral domains. In: Badawi A, Coykendall J (eds) Advances in commutative algebra. Birkhäuser, Singapore, pp 63–92

  6. Azizi A (2003) Weak multiplication modules. Czechoslov Math J 53(3):529–534.

  7. Badawi A (1999) On divided commutative rings. Commun Algebra 27(3):1465–1474.

  8. Beiranvand PK, Beyranvand R (2019) Almost prime and weakly prime submodules. J Algebra Appl 18(7):1950129.

  9. Darani AY, Soheilnia F (2011) 2-absorbing and weakly 2-absorbing submodules. Thai J Math 9(3):577–584

  10. Dobbs D (1976) Divided rings and going-down. Pac J Math 67(2):353–363

  11. Dobbs DE, Shapiro J (2009) A generalization of divided domains and its connection to weak Baer going-down rings. Commun Algebra 37(10):3553–3572.

  12. El-Bast Abd Z, Smith PF (1998) Multiplication modules. Commun Algebra 16(4):755–779.

  13. Koc S, Uregen RN, Tekir U (2017) On 2-absorbing quasi primary submodules. Filomat 31(10):2943–2950.

  14. Koc S, Tekir U, Ulucak G (2019) On strongly quasi primary ideals. Bull Korean Math Soc 56(3):729–743.

  15. MacDonald IG (1973) Secondary representation of modules over a commutative ring. Sympos Math XI:23–43

  16. Matlis E (1960) Divisible modules. Proc Am Math Soc 11(3):385–391

  17. McCasland RL, Moore ME (1992) Prime submodules. Commun Algebra 20(6):1803–1817.

  18. McCasland RL, Moore ME, Smith PF (1997) On the spectrum of a module over a commutative ring. Commun Algebra 25(1):79–103.

  19. Nagata M (1962) Local rings (Interscience tracts in pure and applied mathematics). Wiley, New York

  20. Northcott DG (2008) Lessons on rings, modules and multiplicities. Cambridge University Press, Cambridge

  21. Payrovi SH, Babaei S (2012) On 2-absorbing submodules. Algebra Colloq 19:913–920.

  22. Sharpe T, Sharpe DW, Vámos P (1972) Injective modules. Cambridge University Press, Cambridge

  23. Smith PF (1988) Some remarks on multiplication modules. Arch Math 50(3):223–235.

  24. Wisbauer R (2018) Foundations of module and ring theory. Routledge, Abingdon

  25. Yassemi S (2001) The dual notion of prime submodules. Arch Math 37(4):273–278

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We would like to thank the referee for his/her great effort in proofreading the manuscript.

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Correspondence to Ünsal Tekir.

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Tekir, Ü., Ulucak, G. & Koç, S. On Divided Modules. Iran J Sci Technol Trans Sci 44, 265–272 (2020).

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  • Divided ring
  • Divided module
  • Trivial extension

Mathematics Subject Classification

  • 13C10
  • 13A15
  • 15A03
  • 13F30