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Journals / Hacettepe Journal of Mathematics and Statistics / 2020 / Cilt: 49 - Sayı: 4

Delta operation on modules, prime and radical submodules and primary decomposition

Ashkan Nikseresht
Pages
1303–1314
Publication Date
2022-10-18
DOI
—
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Moreover, we state some conditions under which primary decompositions of submodules of $M$ lift to $M\oplus R$.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:Let $R$ be a commutative ring with identity and $M$ be an $R$-module.
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Key Findings & Data:In this paper, in order to study prime submodules, radical submodules and primary decompositions in finitely generated free $R$-modules, we introduce and study an operation $\Delta: (M\oplus R)^2\to M$ defined by $\Delta(m+r, m'+r')= r'm-rm'$.
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Conclusion & Impact:Moreover, we state some conditions under which primary decompositions of submodules of $M$ lift to $M\oplus R$.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Delta operation · primary decomposition · prime submodule

Abstract

Let $R$ be a commutative ring with identity and $M$ be an $R$-module. In this paper, in order to study prime submodules, radical submodules and primary decompositions in finitely generated free $R$-modules, we introduce and study an operation $\Delta: (M\oplus R)^2\to M$ defined by $\Delta(m+r, m'+r')= r'm-rm'$. In particular, using this operation we give a characterization of prime submodules of $M\oplus R$, in terms of prime submodules of $M$. As an application, we present a characterization of prime submodules of finitely generated free modules. Also we present a formula for the prime radical of submodules of $M\dis R$. Moreover, we state some conditions under which primary decompositions of submodules of $M$ lift to $M\oplus R$.

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Cite This Article

Nikseresht, A. (2022). Delta operation on modules, prime and radical submodules and primary decomposition. Hacettepe Journal of Mathematics and Statistics, Cilt: 49 - Sayı: 4, 1303–1314. https://ajindex.com/articles/529657

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References (22)

  1. M. Alkan and Y. Tıraş, On prime submodules, Rocky Mount. J. Math. 37 (3), 709– 722, 2007
  2. B. Amini and A. Amini, On strongly superfluous submodules, Comm. Algebra 40 (8), 2906–2919, 2012
  3. A. Azizi, Radical formula and prime submodules, J. Algebra 307, 454–460, 2007
  4. A. Azizi, Prime submodules of artinian modules, Taiwanese J. Math. 13 (6B), 2011– 2020, 2009
  5. A. Azizi, Radical formula and weakly prime submodules, Glasgow Math. J. 51, 405– 412, 2009
  6. A. Azizi and A. Nikseresht, Simplified radical formula in modules, Houston J. Math. 38 (2), 333–344, 2012
  7. M. Behboodi and H. Koohi, Weakly prime modules, Vietnam J. Math. 32, 185–195, 2004
  8. S. Çeken and M. Alkan, On Prime submodules and primary decomposition in twogenerated free modules, Taiwanese J. Math. 17 (1), 133–142, 2013
  9. J. Jenkins and P.F. Smith, On the prime radical of a module over a commutative ring, Comm. Algebra 20 (12), 3593–3602, 1992
  10. K.H. Leung and S.H. Man, On commutative Noetherian rings which satisfy the radical formula, Glasgow Math. J. 39, 285–293, 1997
  11. S.H. Man, One dimensional domains which satisfy the radical formula are Dedekind domains, Arch. Math. 66, 276–279, 1996
  12. S.H. Man, On commutative Noetherian rings which satisfy the generalized radical formula, Comm. Algebra 27 (8), 4075–4088, 1999
  13. R. McCasland and M.E. Moore, On radicals of submodules of finitely generated modules, Canad. Math. Bull. 29 (1), 37–39, 1986
  14. R.L. McCasland and P.F. Smith, Zariski spaces of modules over arbitrary rings, Comm. Algebra 34, 3961–3973, 2006
  15. F. Mirzaei and R. Nekooei, On prime submodules of a finitely generated free module over a commutative ring, Comm. Algebra 44 (9), 3966–3975, 2016
  16. M.E. Moore and S.J. Smith, Prime and radical submodules of modules over commutative rings, Comm. Algebra 30 (10), 5037–5064, 2002
  17. A. Nikseresht and A. Azizi, On radical formula in modules, Glasgow Math. J. 53, 657–668, 2011
  18. A. Nikseresht and A. Azizi, Envelope dimension of modules and the simplified radical formula, Canad. Math. Bull. 56 (4), 683–694, 2013
  19. A. Parkash, Arithmetical rings satisfy the radical formula, J. Commut. Algebra 4 (2), 293–296, 2012
  20. D. Pusat-Yilmaz and P. F. Smith, Modules which satisfy the radical formula, Acta. Math. Hungar. 95, 155-167, 2002
  21. P. F. Smith, Primary modules over commutative rings, Glasgow Math. J. 43, 103–111, 2001
  22. Y. Tıraş and M. Alkan, Prime modules and submodules, Comm.