Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 14 (2019), 61 -- 108

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SURVEY ON COMULTIPLICATION MODULES

Habibollah Ansari-Toroghy and Faranak Farshadifar

Abstract. The concept of comultiplication (as a dual notion of multiplication) modules was introduced and studied by H. Ansari-Toroghy and F. Farshadifar in 2007. This notion has obtained a great attention by many authors and now there is a considerable amount of research concerning this class of modules. The main purpose of this paper is to collect these results and provide a useful source for those who are interested in research in this field.

2010 Mathematics Subject Classification: 13C13; 13C99.
Keywords: Comultiplication module; completely irreducible submodule; finitely generated; Goldie dimension; complements; endomorphism; strongly comultiplication module; copure submodule; cocyclic module; P-cotorsion module; P-cocyclic module; Fitting ideal; graded comultiplication module; fully coidempotent module; second submodule; weak comultiplication module.

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References

  1. R. Abu-Dawwas and M. Ali, Comultiplication modules over strongly graded rings, Inte. J. Pure and Applied Mathematics, 81 (5) (2012), 693-699.

  2. J. Y. Abuhlail, Zariski topologies for coprime and second submodules, Algebra Colloq., 22 (47) (2015), 47-72. MR3296756. Zbl 1312.16002.

  3. S.~Abu-Saymeh, On dimension of finitely generated modules, Comm. Algebra 23 (3) (1995), 1131--1144. ‎MR1316753‎.

  4. M.M. Ali, Idempotent and nilpotent submodules of multiplication modules, Comm. Algebra, 36 (12) (2008), 4620-4642. ‎MR2473351. Zbl 1160.13004.

  5. M. M. Ali and D. J. Smith, Pure submodules of multiplication modules, Beiträge Algebra Geom., 45 (1) (2004), 61-74. MR2070633. Zbl 1097.13014. ‎

  6. Y.~ Al-Shaniafi and P. F. ~Smith, Comultiplication modules over commutative rings, J. Commut. Algebra, 3 (1) (2011), 1-29. MR2782697. Zbl 1238.13018.

  7. Y.~ Al-Shaniafi and P. F. ~Smith, Comultiplication modules over commutative rings II, J. Commut. Algebra, 4 (2) (2012), 153-174. ‎MR2959013‎. Zbl 1260.13017.

  8. W. ~Anderson and K.R. ~Fuller, Rings and categories of modules, Springer-Verlag, New York-Heidelberg-Berlin, 1974. ‎ MR0417223. Zbl 0301.16001.

  9. H.~Ansari-Toroghy and F.~Farshadifar, Product and dual product of submodules, Far East J. Math. Sci. 25 (3) (2007), 447-455. ‎MR2335105‎. Zbl 1133.13010.

  10. H.~Ansari-Toroghy and F.~Farshadifar, The dual notion of multiplication modules, Taiwanese J. Math. 11 (4) (2007), 1189--1201. MR2348561. Zbl 1137.16302.

  11. H.~Ansari-Toroghy and F.~Farshadifar, On endomorphisms of multiplication and comultiplication modules, Arch. Math. (Brno) 44 (2008), 9--15. ‎MR2431226‎. Zbl 1212.13006.

  12. H.~Ansari-Toroghy and F.~Farshadifar, Comultiplication modules and related results, Honam Math. J. 30 (1) (2008), 91--99. ‎MR2399778. Zbl 1180.13016.

  13. H.~Ansari-Toroghy and F.~Farshadifar, On comultiplication modules, Korean Ann. Math. 25 (2) (2008), 57--66. ‎MR2475337‎.

  14. H.~Ansari-Toroghy and F.~Farshadifar, Strong comultiplication modules, CMU. J. Nat. Sci. 8 (1) (2009), 105--113.

  15. H.~Ansari-Toroghy and F.~Farshadifar, On the dual notion of multiplication modules, AJSE-Mathematics, 36 (6) (2011), 925-932. ‎ MR2845520‎.

  16. H.~Ansari-Toroghy and F.~Farshadifar, Graded comultiplication modules, Chiang Mai. J. Sci. 38 (2) (2011), 1-10. ‎MR2832354‎. Zbl 1231.16035.

  17. H.~Ansari-Toroghy and F.~Farshadifar, On multiplication and comultiplication modules, Acta Math. Sci. 31B (2) (2011), 694-700. ‎MR2817127‎. Zbl 1240.13015.

  18. H. Ansari-Toroghy and F. Farshadifar, Multiplication and comultiplication modules, Novi Sad J. Math. 41 (2) (2011), 117-122. MR2933042. Zbl 1265.13011. ‎

  19. H.~Ansari-Toroghy and F.~Farshadifar, On the dual notion of prime submodules, Algebra Colloq. 19 (Spec 1) (2012), 1109-1116. MR3073398‎. Zbl 1294.13012. ‎

  20. H.~Ansari-Toroghy and F.~Farshadifar, On the dual notion of prime submodules (II), Mediterr. J. Math. 9 (2) (2012), 329-338. MR2917107. Zbl 1253.13013. ‎

  21. H. Ansari-Toroghy and F. Farshadifar, Fully idempotent and coidempotent modules, Bull. Iranian Math. Soc. 38 (4 ) (2012), 987-1005. ‎MR3028698. Zbl 1311.13010.

  22. H.~Ansari-Toroghy and F.~Farshadifar, On the dual notion of prime radicals of submodules, Asian Eur. J. Math. 6 (2) (2013), 1350024 (11 pages). ‎MR3091647‎. Zbl 1278.13012.

  23. H.~Ansari-Toroghy and F.~Farshadifar, The Zariski topology on the second spectrum of a module, Algebra Colloq. 21 (04 ) (2014), 671-688. MR3266519‎. Zbl 1309.13011. ‎

  24. H. Ansari-Toroghy and F. Farshadifar, The dual notion of some generalizations of prime submodules, Comm. Algebra, 39 (2011), 2396-2416. ‎ MR2821719‎.

  25. H. Ansari-Toroghy, F. Farshadifar, and M. Mast-Zohouri, Some remarks on multiplication and comultiplication modules, Inte. Math. Forum, 4 (6) (2009), 287-291. ‎MR2491250. Zbl 1169.13303.

  26. H. Ansari-Toroghy, S. Keyvani, and F. Farshadifar, The Zariski topology on the second spectrum of a module (II), Bull. Malays. Math. Sci. Soc. (2). 39 (3) (2016), 1089-1103. MR3515068. Zbl 1348.13019. ‎

  27. H. Ansari-Toroghy and F. Mahboobi-Abkenar, The large sum graph related to comultiplication modules, Le Mathematiche, 73 (1)(2018), 59-69. MR3815387. Zbl 1402.13008. ‎

  28. J. A’zami and M. Khajepour, Comultiplication Modules, Eur. J. Pure Appl. Math. 9(3) (2016), 244-249. ‎MR3522834‎. Zbl 1359.13009.

  29. A.~Barnard, Multiplication modules, J. Algebra 71 (1981), 174--178. ‎MR0627431‎. Zbl 0468.13011.

  30. F. Callialp, U.Tekir, and G. Ulucak, Comultiplication lattice modules, IJST, 39 (2) (2015), 213-220. ‎MR3511976.

  31. S. Ceken, M. Alkan, and P. F. Smith, Second modules over noncommutative rings, Comm. Algebra, 41 (1) (2013), 83-98. MR3010523. Zbl 1270.16001.

  32. S. Ceken, M. Alkan, and P. F. Smith, The dual notion of the prime radical of a module, J. Algebra 392 (2013), 265-275. MR3085034. Zbl 1301.16001. ‎

  33. C. W. Choi and P. F. Smith, On endomorphisms of multiplication modules, J. Korean Math. Soc. 31 (1) (1994), 89-95. MR1269453. Zbl 0820.13003. ‎

  34. J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting modules, Birkhauser Verlag, Basel, 2006. MR2253001‎. Zbl 1102.16001. ‎

  35. J.~Dauns, Prime submodules, J. Reine Angew. Math. 298 (1978), 156--181. ‎MR0498715‎.

  36. J.~Dauns, Primal modules. Commun. Algebra 25 (8) (1997), 2409–2435. MR1459569. Zbl 0882.16001. ‎

  37. R. Ebrahimi Atani and Sh. Ebrahimi Atani, Comultiplication modules over a pullback of Dedekind domains, Czechoslovak Math. J., 59 (4) (2009), 1103-1114. MR2563581‎. Zbl 1224.13012. ‎

  38. D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Springer-verlag, New York 1995. MR1322960‎. Zbl 1224.13012. ‎

  39. Z.A. El-Bast and P.F. Smith, Multiplication modules, Comm. Algebra 16 (1988), 755--779. ‎MR0932633. Zbl 0642.13002.

  40. C. Faith, Algebra II: Ring Theory, Springer-Verlag, New York-Heidelberg-Berlin, 1976. ‎MR0427349‎. Zbl 0335.16002.

  41. C. Faith, Rings whose modules have maximal submodules, Publ. Mat. 39 (1995), 201-214. ‎MR1336364‎. Zbl 0840.16002.

  42. D.J. Fieldhouse, Semi-perfect and F-semi-perfect modules, Int. J. Math. Math. Sci.8 (3) (1985), 545-548. ‎MR0809075‎. Zbl 0581.16019.

  43. H. Fitting, Die Determinantenideale eines Moduls, Jahresbericht d. Deutschen Math.-Vereinigung, 46 (1936), 195-228. Zbl 0016.05003.

  44. L.~Fuchs, W.~Heinzer and B.~Olberding, Commutative ideal theory without finiteness conditions: Irreducibility in the quotient filed, in: Abelian Groups, Rings, Modules, and Homological Algebra, Lect. Notes Pure Appl. Math. 249 (2006), 121--145. ‎MR2229107‎. Zbl 1098.13003.

  45. A. Ghorbania and A. Haghang, Generalized Hopfian modules, J. Algebra 255 (2002), 324–341. ‎MR1935502‎. Zbl 1014.16007.

  46. A. Haghang and M. R. Vedali, Modules whose injective endomorphism are essential, J. Algebra 243 (2001), 765-779. ‎ MR1850657.

  47. J. Hausen and J. A. Johnson, Centralizer near rings that are rings. J. Austral Soc (Series A), 59 (1995), 173–183. ‎MR1346625. Zbl 0852.16032.

  48. S. Karimzadeh and S. Hadjirezaei, On the fitting ideals of a comultiplication modules, Journal of Algebra and Related Topics, 3 (1) (2015), 31-39. ‎MR3439524. Zbl 1334.13005.

  49. W. Krull, Uber Multiplikationsringe, Sitzber. Heidelberg. Akad. Wiss. Abhandl., 5 (1925), 13-18. JFM 51.0120.02.

  50. W. Krull, Uber allgemeine multiplications rings, Tohoku Math. J. 41 (1936), 320-326. JFM 62.0095.02.

  51. D. K. Tütüncü, When is a fully idempotent module a V-module, Bull. Math. Soc. Sci. Math. Roumanie, Tome, 53(101)(4) (2010) 387-391. ‎MR2777682‎. Zbl 1219.16010.

  52. H.~Matsumara, Commutative Ring Theory, Cambridge University Press, Cambridge 1986. Zbl 0603.13001.

  53. R.L. McCasland and M. E. Moore, On radical of submodules of finitely generated modules, Canad. Math. Bull. 29(1) (1986), 37-39. ‎MR0824879. Zbl 0546.13001.

  54. J. R. Munkres, Topology: A First Course. Prentice Hall, 1975. Zbl 0306.54001. ‎MR3728284.

  55. C. Nastasescu and F. Van Oystaeyen, Graded ring theory, Mathematical Library 28, North Holand, Amsterdam, 1982. ‎MR0676974‎. Zbl 0494.16001.

  56. A. Nikseresht and H. Sharif, On comultiplication and r-multiplication modules, Journal of Algebraic Systems, 2 (1) (2014), 1-19.

  57. N. Phadatare, S. Ballal, and V. Kharat, On the second spectrum of lattice modules, Discussiones Mathematicae-General Algebra and Applications, 37 (1) (2017), 59-74. ‎ MR3648183‎.

  58. S. Rajaee, Some results on comultiplication modules, Inte. J. Algebra, 7 (7) (2013), 337 - 340. MR3056458. Zbl 1284.13011. ‎

  59. E. Rostami, S. Karimzadeh, and R. Nekooei, Decomposition of modules into modules whose annihilators are pseudo-irreducible, Communications in Algebra, DOI: 10.1080/00927872.2018.1448849. MR3864253. Zbl 06959284.‎

  60. J. K. Sahani, Some more remarks on comultiplication modules, International Journal of Science and Research, 3 (6) (2014), 1839-1843.

  61. D.W. Sharpe and P.~Vamos, Injective modules, Cambridge University Press, 1972. ‎MR0360706. Zbl 0245.13001.

  62. R. Y. Sharp, Step in commutative algebra, Cambridge University Press, 1990. ‎MR1070568‎. Zbl 0703.13001.

  63. P.F. Smith, Some remarks on multiplication modules, Arch. Math. 50 (1988), 223--235. ‎MR0933916‎. Zbl 0615.13003.

  64. A.A. Tuganbaev, Semidistributive modules, Journal of Mathematical Sciences, 94 (6) (1999), 1809-1887. ‎MR1703199. Zbl 0936.16007.

  65. A.A. Tuganbaev, Comultiplication modules over noncommutative rings, Journal of Mathematical Sciences, 191 (5) (2013), 743-747. MR2954691‎. Zbl 1280.16006. ‎

  66. Yongduo Wang and Yang Liu, A note on comultiplication modules, Algebra Colloq. 21( 01) (2014), 147-150. ‎MR3163368. Zbl 1282.13020.

  67. R.~Wisbauer, Foundations of Modules and Rings Theory, Gordon and Breach, Philadelphia, PA, 1991. ‎MR1144522. Zbl 0746.16001.

  68. S.~Yassemi, The dual notion of prime submodules, Arch. Math (Brno) 37 (2001), 273--278. ‎MR1879449. Zbl 1090.13005.

  69. S.~Yassemi, Coassociated primes, Comm. Algebra. 23 (1995), 1473--1498. MR1317409. Zbl 0832.13004. ‎


Habibollah Ansari-Toroghy
Department of pure Mathematics, Faculty of Mathematical Sciences,
University of Guilan,
P. O. Box 41335-19141, Rasht, Iran.
e-mail: ansari@guilan.ac.ir

Faranak Farshadifar (Corresponding Author)
Department of Mathematics, Farhangian University,
Tehran, Iran.
e-mail: f.farshadifar@cfu.ac.ir

http://www.utgjiu.ro/math/sma