Pendant and isolated vertices of comaximal graphs of modules
Résumé
A comaximal graph Γ(M) is an undirected graph with vertex set as the collection of all submodules of a module M and any two vertices A and B are adjacent if and only if A + B = M. We discuss characteristics of pendant vertices in Γ(M). We also observe features of isolated vertices in a special spanning subgraph in Γ(M).
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Références
Akbari, S., Habibi, M., Majidinya, A., Manaviyat, R.: A note on comaximal graph of non-commutative rings, Algebras and Representation Theory, 16 (2) (2013) 303-307. DOI: https://doi.org/10.1007/s10468-011-9309-z
Anderson, D. F., Badawi, A.: The total graph of a commutative ring, J. of Algebra, 320 (2008) 2706–2719. DOI: https://doi.org/10.1016/j.jalgebra.2008.06.028
Ashrafi, N.; Maimani, H.R.; Pournaki, M.R.; Yassemi, S.: Unit graphs associated with rings, Comm. Algebra, 38 (8) (2010) 2851–2871. DOI: https://doi.org/10.1080/00927870903095574
Atiyah, M.F., Macdonald, I.G.: Introduction to commutative algebra, Addison-Wesley, London, 1969.
Anderson, F. W., Fuller, K. R.: Rings and Categories of Modules, Managing Editors P. R. Halmos, 1978.
Barnard, A.: Multiplication Modules, J. Algebra, 71 (1981) 174-178. DOI: https://doi.org/10.1016/0021-8693(81)90112-5
Balkrishnan, R., Ranganathan, K.: A text book of graph theory, Springer-verlag New York, Inc. Reprint, 2008.
Beck, I.: Coloring of commutative rings, J. Algebra, 116 (1988) 208-226. DOI: https://doi.org/10.1016/0021-8693(88)90202-5
Chakrabarty, I., Ghosh, S., Mukherjee, T.K., Sen, M.K.: Intersection graphs of ideals of rings, Discrete Math., 309 (17) (2009) 5381–5392. DOI: https://doi.org/10.1016/j.disc.2008.11.034
Harary, F.: Graph theory, Addison-Wesley Publishing Company, Inc., Reading, Mass., 1969.
Huckaba, J. A.: Commutative rings with zero-divisors, Marcel-Dekker, New York, Basel, 1988.
Lambeck, J.: Lectures on rings and modules, Blaisdell Publishing Company, Waltham, Toronto, London, 1966.
Kasch, F.: Modules and rings, Academic Press Inc. (London) Ltd., 1982.
Kaplansky, I.: Commutative rings, revised Edition, University of Chicago Press, Chicago, 1974.
Kirby , D.: Closure operations on ideal and submodules, J. London Math. Soc., 44 (1969) 283-291. DOI: https://doi.org/10.1112/jlms/s1-44.1.283
Maimani, H.R., Salimi, M., Sattari, A., Yassemi, S.: Comaximal graph of commutative rings, J. Algebra, 319 (2008) 1801-1808. DOI: https://doi.org/10.1016/j.jalgebra.2007.02.003
Mijbass, A. S., Abdullah, N. K.: Semi-small submodules, Tikrit Journal of Pure Science, 16 (1) (2011) 104-107.
Naoum , A. G.: On the ring of endomorphisms of finitely generated multiplication modules , Periodica Mathematica Hungarica, 29 (1994) 277-284. DOI: https://doi.org/10.1007/BF01875854
Sharma, P. K.; Bhatwadekar, S.M.: A note on graphical representation of rings, J. Algebra, 176 (1) (1995) 124–127. DOI: https://doi.org/10.1006/jabr.1995.1236
Wang, H.: Graphs associated to co-maximal ideals of commutative rings, J. Algebra, 320 (2008) 2917-2933. DOI: https://doi.org/10.1016/j.jalgebra.2008.06.020
Wisbauer, R., Foundations of module and ring theory, Gordon and Breach Science Publishers, Reading, 1991.
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