Free and cyclic canonical $\bf{(m,n)-}$ ary hypermodules

Authors

  • Z. Belali Department ofMathematics, Yazd University, Yazd, Iran.
  • S. M. Anvariyeh Department ofMathematics, Yazd University, Yazd, Iran.
  • S. Mirvakili Mathematics Department, Payame Noor University, 19395-4697 Tehran, I. R. of IRAN.

DOI:

https://doi.org/10.5556/j.tkjm.42.2011.870

Keywords:

Canonical $m-$ary hypergroup, Krasner $(m, n)-$ary hyperring, free $(m, n)-$ary hypermodules.

Abstract

In this paper, the class of free and cyclic canonical $(m,n)-$ary hypermodules over Krasner $(m,n)-$ary hyperrings is defined. Free canonical $(m,n)-$ary hypermodules are a generalization of free canonical hypermodules and a generalization of free modules. Also, several properties are found. In addition, we introduce the concepts of a free basis and a free $(m,n)$-hypermodules as a free object in the category of $(m,n)$-hypermodules and prove some results in this respect. Finally, we obtain some results and relations among a finitely generated torsion free and a free $(m,n)$-hypermodule.

References

S. M. Anvariyeh, S. Mirvakili and B. Davvaz, $theta^ast$-relation on hypermodules and fundamental modules over commutative fundamental rings Communications in Algebra, 36 (2008), 622–631.

S. M. Anvariyeh, B. Davvaz, Strongly transitive geometric spaces associated to hypermodules, Journal of Algebra, 322 (2009), 1340–1359.

S. M. Anvariyeh, S. Mirvakili and B. Davvaz, Transitivity of $theta^ast$−relation on hypermodules, Iranian J. Science Tech., Transaction A, 32(A3) (2008), 188–205.

S. M. Anvariyeh, S. Mirvakili and B. Davvaz, Fundamental relation on (m,n)−ary hypermodules over (m,n)−ary hyperrings, Ars Combinatoria, 94 (2010), 273–288.

G. Crombez, On (n,m)-rings, Abh.Math. Semin. Univ. Hamburg, 37 (1972), 180–199.

G. Crombez and J. Timm, On (n,m)-quotient rings, Abh.Math. Semin. Univ. Hamburg, 37 (1972), 200–203.

B. Davvaz and T. Vougiouklis, N-ary hypergroups , Iranian Journal of scince and Technology, Transaction A,

(A2) (2006), 165-174.

W. Dörnte,Untersuchungen über einen verallgemeinerten Gruppenbegriff,Math. Z., 29 (1928), 1–19.

W. A. Dudek, Remarks on n-ary groups, DemonstratioMath., 13 (1980), 165–181.

W. A. Dudek, Idempotents in n-ary semigroups, Southeast Asian Bull.Math., 25 (2001), 97–104.

W. A. Dudek, Varieties of polyadic groups, Fiiomat, 9 (1995), 557-674.

M. Ghadiri and B. N. Waphare, n−ary polygroup, Iranian Journal of Scince and Technology, Transaction A,

Vol.32, No.A2.

M. Krasner, A class of hyperrings and hyperfields, Lnt. J.Math. &Math. Sci., 6 (1983), 307–312.

V. Leoreanu- Fotea, Canonical n−ary hypergroup, Italian J. Pure Appl.Math., 24 (2008).

V. Leoreanu- Fotea, n−hypergroups and binary relations, European Journal of Combinatorics, 29 (2008), 1207–1218.

F.Marty, Sur une generalization de la notion de groupe, 8iem congresMath. Scandinaves, Stockholm, (1934), 45–49.

C. G. Massouros, Free and cyclic hypermodules, Annali diMatematica pura ed applicata (IV), Vol. Cl, (1988), 153–166.

S.Mirvakili and B. Davvaz, Relations on Krasner (m,n)−hyperrings, European Journal of Combinatorics, European Journal of Combinatorics, 31(2010), 790-802.

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Published

2011-03-01

How to Cite

Belali, Z., Anvariyeh, S. M., & Mirvakili, S. (2011). Free and cyclic canonical $\bf{(m,n)-}$ ary hypermodules. Tamkang Journal of Mathematics, 42(1), 105–118. https://doi.org/10.5556/j.tkjm.42.2011.870

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Papers