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Functors , , and in the Category of A - Alg

Received: 18 February 2019     Accepted: 26 March 2019     Published: 18 April 2019
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Abstract

The purpose of this paper is to study some results of homological algebra in the category A-Alg (resp. Alg-A) of left (resp. right) A-algebra in the noncommutative case. In this paper A is a subring of B. So the main results of this paper are, if B is a noetherian duo-ring, S a central saturated multiplicatively closed subset of A, SR the set of regular elements of S, a finitely presented right A-algebra and a (B-A)-bialgebra, then is isomorphic to , also is isomorphic to , for any integer n.

Published in International Journal of Theoretical and Applied Mathematics (Volume 5, Issue 1)
DOI 10.11648/j.ijtam.20190501.11
Page(s) 1-9
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2019. Published by Science Publishing Group

Keywords

Algebra, Multiplicatively Closed Subset, Ore Condition of Multiplicatively Closed Subset, Localized, Category, Functor, Complex Projective Resolution

References
[1] F. W. Anderson Kent R.Fuller, Rings and Categories of Modules, Springer-Verlag New York, 1974, 1992 Inc.
[2] M. F. Atiyah and I.G.Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Company, University of Oxford.
[3] H. Cartan and Samuel Eilenberg, Homological Agebra, Princeton University Press, New Jersey, 1956.
[4] F. Dennis, Noncommutative algebra, GTM Vol.144, Springer-Verlag, 1993.
[5] M. P. Eelbert, Localisation in Duo-ring, Kansas.Ciry, Missouri.
[6] M. F. Maaouia and al, Functor S-1 () and Adjoint Isomorphism, International Mathematical Forum, vol. 11, 2016, no. 5, 227-237.
[7] N. Jacobson, Structure of rings, Amer.Math.Soc.Colloquium Publications, Vol.37, 1956.
[8] D. M. Kan, Adjoint functors, Trans. Amer. Math. Soc. 87 (1958), 294-329.
[9] Lam, A First Course in Noncommutative Rings, GTM Vol.131, Springer-Verlag, 2001.
[10] M. F. Maaouia and al., Localization in a Duo-Ring and Polynomials Algebra, Springer International Publishing Switzerland 2016 C.T. Gueye, M.S. Molina (eds.), Non-Associative and Non-Commutative Algebra and Operator Theory, Springer Proceedings in Mathematics and Statistics 160, DOI 10.1007 978-3-319-32902-4-13.
[11] M. F. Maaouia, Thèse d’état, Faculté des Sciences et Techniques, UCAD, Dakar, Juillet, 2011.
[12] M. F. Maaouia, Doctorat 3ème Cycle, Faculté des Sciences et Techniques, UCAD, Dakar, Juillet, 2003.
[13] M. F. Maaouia and M.Sanghare, Localisation Dans Les Duo-Anneaux, Afrika Mathematika, 2009.
[14] M. F. Maaouia and M.Sanghare, Module de fraction- Sous-modules S-saturée et foncteur S-1, International Journal of Algebra, 16(2012), 0973-1768.
[15] M. F. Maaouia and M.Sanghare, Anneau de valuation non nécessairement commutatif et duo-anneau de Dideking, Global Journal of Pure and Applied Mathematics, ISSN 0973-1768 Volume 8, Number 1(2012).
[16] H. Matsumura, Commutative algebra, W.A. Benjamin, New-York, 1970.
[17] R. Pierce, Associative Algebras, Grad.Text in Math.88, Springer, 1982.
[18] J. Rotman, Notes on Homological Algebra, University of Illinois, Urbane, 1968.
[19] J. Rotman, An introduction to Homological Algebra, GSM Vol.114, Academic Press, New York 1972.
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  • APA Style

    Moussa Thiaw, Mohamed Ben Faraj Ben Maaouia, Mamadou Sanghare. (2019). Functors , , and in the Category of A - Alg. International Journal of Theoretical and Applied Mathematics, 5(1), 1-9. https://doi.org/10.11648/j.ijtam.20190501.11

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    ACS Style

    Moussa Thiaw; Mohamed Ben Faraj Ben Maaouia; Mamadou Sanghare. Functors , , and in the Category of A - Alg. Int. J. Theor. Appl. Math. 2019, 5(1), 1-9. doi: 10.11648/j.ijtam.20190501.11

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    AMA Style

    Moussa Thiaw, Mohamed Ben Faraj Ben Maaouia, Mamadou Sanghare. Functors , , and in the Category of A - Alg. Int J Theor Appl Math. 2019;5(1):1-9. doi: 10.11648/j.ijtam.20190501.11

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  • @article{10.11648/j.ijtam.20190501.11,
      author = {Moussa Thiaw and Mohamed Ben Faraj Ben Maaouia and Mamadou Sanghare},
      title = {Functors , ,  and in the Category of A - Alg},
      journal = {International Journal of Theoretical and Applied Mathematics},
      volume = {5},
      number = {1},
      pages = {1-9},
      doi = {10.11648/j.ijtam.20190501.11},
      url = {https://doi.org/10.11648/j.ijtam.20190501.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20190501.11},
      abstract = {The purpose of this paper is to study some results of homological algebra in the category A-Alg (resp. Alg-A) of left (resp. right) A-algebra in the noncommutative case. In this paper A is a subring of B. So the main results of this paper are, if B is a noetherian duo-ring, S a central saturated multiplicatively closed subset of A, SR the set of regular elements of S,  a finitely presented right A-algebra and  a (B-A)-bialgebra, then  is isomorphic to , also  is isomorphic to , for any integer n.},
     year = {2019}
    }
    

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    T1  - Functors , ,  and in the Category of A - Alg
    AU  - Moussa Thiaw
    AU  - Mohamed Ben Faraj Ben Maaouia
    AU  - Mamadou Sanghare
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    DO  - 10.11648/j.ijtam.20190501.11
    T2  - International Journal of Theoretical and Applied Mathematics
    JF  - International Journal of Theoretical and Applied Mathematics
    JO  - International Journal of Theoretical and Applied Mathematics
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    PB  - Science Publishing Group
    SN  - 2575-5080
    UR  - https://doi.org/10.11648/j.ijtam.20190501.11
    AB  - The purpose of this paper is to study some results of homological algebra in the category A-Alg (resp. Alg-A) of left (resp. right) A-algebra in the noncommutative case. In this paper A is a subring of B. So the main results of this paper are, if B is a noetherian duo-ring, S a central saturated multiplicatively closed subset of A, SR the set of regular elements of S,  a finitely presented right A-algebra and  a (B-A)-bialgebra, then  is isomorphic to , also  is isomorphic to , for any integer n.
    VL  - 5
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Author Information
  • UFR of Applied Sciences and Technology, Gaston Berger University, Saint-Louis, Senegal

  • UFR of Applied Sciences and Technology, Gaston Berger University, Saint-Louis, Senegal

  • Department of Mathematics, Faculty of Science and Technology, Cheikh Anta Diop University, Dakar, Senegal

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