Jaime CASTRO PÉREZ, Mauricio Gabriel MEDINA-BÁRCENAS

72110

Johns modules and quasi-Johns modules

A right Johns ring is a right Noetherian ring in which every right ideal is a right annihilator. It is known that in a Johns ring RR the Jacobson radical J(R)J(R) of RR is nilpotent and Soc(R)(R) is an essential right ideal of RR. Moreover, every right Johns ring RR is right Kasch, that is, every simple right RR-module can be embedded in RR. For a M∈RM∈R-Mod we use the concept of MM-annihilator and define a Johns module (resp. quasi-Johns) as a Noetherian module MM such that every submodule is an MM-annihilator. A module MM is called quasi-Johns if any essential submodule of MM is an MM-annihilator and the set of essential submodules of MM satisfies the ascending chain condition. In this paper we extend classical results on Johns rings, as those mentioned above and we also provide new ones. We investigate when a Johns module is Artinian and we give some information about its prime submodules.
Keywords:

Johns module, Noetherian module, M-annihlator, Kasch module (semi)prime submodule, projective module,

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  • [1] T. Albu and R. Wisbauer, Kasch modules, in: Advances in Ring Theory, pages 1–16. Springer, 1997.
  • [2] I. Assem, A. Skowronski, and D. Simson, Elements of the Representation Theory of Associative Algebras: Volume 1: Techniques of Representation Theory, volume 65, Cambridge University Press, 2006.
  • [3] J. Beachy, M-injective modules and prime M-ideals, Comm. Algebra, 30 (10), 4649– 4676, 2002.
  • [4] J. Castro Pérez, M. Medina Bárcenas, and J. Ríos Montes, Modules with ascending chain condition on annihilators and Goldie modules, Comm. Algebra, 45 (6), 2334– 2349, 2017.
  • [5] J. Castro Pérez, M. Medina Bárcenas, J. Ríos Montes, and A. Zaldívar Corichi, On semiprime Goldie modules, Comm. Algebra, 44 (11), 4749–4768, 2016.
  • [6] J. Castro Pérez, M. Medina Bárcenas, J. Ríos Montes, and A. Zaldívar Corichi, On the structure of Goldie modules, Comm. Algebra, 46 (7), 3112–3126, 2018.
  • [7] J. Castro Pérez and J. Ríos Montes, FBN modules, Comm. Algebra, 40 (12), 4604– 4616, 2012.
  • [8] J. Castro Pérez and J. Ríos Montes, Prime submodules and local Gabriel correspondence in $\sigma[{M}]$, Comm. Algebra, 40 (1), 213–232, 2012.
  • [9] N.V. Dung, D. Van Huynh, P.F. Smith, and R. Wisbauer, Extending modules, volume 313, CRC Press, 1994.
  • [10] C. Faith and P. Menal, A counter example to a conjecture of Johns, Proc. Amer. Math. Soc. 116 (1), 21–26, 1992.
  • [11] C. Faith and P. Menal, The structure of Johns rings, Proc. Amer. Math. Soc. 120 (4), 1071–1081, 1994.
  • [12] C. Hajarnavis and N. Norton, On dual rings and their modules, J. Algebra, 93 (2), 253–266, 1985.
  • [13] B. Johns, Annihilator conditions in noetherian rings, J. Algebra, 49 (1), 222–224, 1977.
  • [14] M. Medina-Bárcenas and A.Ç. Özcan, Primitive submodules, co-semisimple and regular modules, Taiwanese J. Math. 22 (3), 545–565, 2018.
  • [15] A.Ç. Özcan, A. Harmanci, and P. Smith, Duo modules, Glasg. Math. J. 48 (3), 533– 545, 2006.
  • [16] F. Raggi, J. Ríos, H. Rincón, R. Fernández-Alonso, and C. Signoret, Prime and irreducible preradicals, J. Algebra Appl. 4 (4), 451–466, 2005.
  • [17] F. Raggi, J. Ríos, H. Rincón, R. Fernández-Alonso, and C. Signoret, Semiprime preradicals, Comm. Algebra, 37 (8), 2811–2822, 2009.
  • [18] L. Shen, A note on quasi-johns rings, in: Contemporary Ring Theory 2011, pages 89–96. World Scientific, 2012.
  • [19] R. Wisbauer, Foundations of module and ring theory, volume 3, Reading: Gordon and Breach, 1991.
  • [20] R. Wisbauer, Modules and Algebras: Bimodule Structure on Group Actions and Algebras, volume 81, CRC Press, 1996.
Hacettepe Journal of Mathematics and Statistics-Cover
  • Yayın Aralığı: 6
  • Başlangıç: 2002
  • Yayıncı: Hacettepe Üniversitesi Fen Fakultesi
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