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Year 2020, Volume: 49 Issue: 4, 1303 - 1314, 06.08.2020
https://doi.org/10.15672/hujms.667410

Abstract

References

  • [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. Algebra 31 (11), 5263–5261, 2003.

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

Year 2020, Volume: 49 Issue: 4, 1303 - 1314, 06.08.2020
https://doi.org/10.15672/hujms.667410

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 $M\oplus R$.

References

  • [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. Algebra 31 (11), 5263–5261, 2003.
There are 22 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Ashkan Nikseresht 0000-0002-4422-8782

Publication Date August 6, 2020
Published in Issue Year 2020 Volume: 49 Issue: 4

Cite

APA Nikseresht, A. (2020). Delta operation on modules, prime and radical submodules and primary decomposition. Hacettepe Journal of Mathematics and Statistics, 49(4), 1303-1314. https://doi.org/10.15672/hujms.667410
AMA Nikseresht A. Delta operation on modules, prime and radical submodules and primary decomposition. Hacettepe Journal of Mathematics and Statistics. August 2020;49(4):1303-1314. doi:10.15672/hujms.667410
Chicago Nikseresht, Ashkan. “Delta Operation on Modules, Prime and Radical Submodules and Primary Decomposition”. Hacettepe Journal of Mathematics and Statistics 49, no. 4 (August 2020): 1303-14. https://doi.org/10.15672/hujms.667410.
EndNote Nikseresht A (August 1, 2020) Delta operation on modules, prime and radical submodules and primary decomposition. Hacettepe Journal of Mathematics and Statistics 49 4 1303–1314.
IEEE A. Nikseresht, “Delta operation on modules, prime and radical submodules and primary decomposition”, Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 4, pp. 1303–1314, 2020, doi: 10.15672/hujms.667410.
ISNAD Nikseresht, Ashkan. “Delta Operation on Modules, Prime and Radical Submodules and Primary Decomposition”. Hacettepe Journal of Mathematics and Statistics 49/4 (August 2020), 1303-1314. https://doi.org/10.15672/hujms.667410.
JAMA Nikseresht A. Delta operation on modules, prime and radical submodules and primary decomposition. Hacettepe Journal of Mathematics and Statistics. 2020;49:1303–1314.
MLA Nikseresht, Ashkan. “Delta Operation on Modules, Prime and Radical Submodules and Primary Decomposition”. Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 4, 2020, pp. 1303-14, doi:10.15672/hujms.667410.
Vancouver Nikseresht A. Delta operation on modules, prime and radical submodules and primary decomposition. Hacettepe Journal of Mathematics and Statistics. 2020;49(4):1303-14.