Hyperrings which every element is sum of an idempotent and nilpotent

Document Type : Research Paper

Authors

Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran

10.29252/as.2019.1360

Abstract

In this paper, we define a generalized of clean of Krasner hyperrings for general hyperrings based on the notation of nilpotent elements of a general hyperring $R$, named nil clean hyperring. We examine characterization of this kind of hyperrings and finally, we obtain some relations of nil clean general hyperrings with other hyperrings.

Keywords

References

[1] R. Ameri, M. Norouzi, New fundamental relation of hyperrings, European J. Combin., 34 (2013) 884-891.
[2] R. Ameri, M. Norouzi, On commutative hyperring, int. Journal of Algebraic Hyperstructures and its Applications, 1 (2014), No. 1, 45-58.
[3] R. Ameri, M. Norouzi, On multiplication (m; n)-hypermodules, European Journal of Combinatorics, 44
(2015), 153-171.
[4] T. Amouzegar, Y. Talebi, On clean hyperrings, Journal of Hyperstructures, 4 (1) (2015), 1-10.
[5] H. Chen, On strongly J-clean rings, Comm. Algebra 38 (2010), 3790-3804.
[6] P. Corsini, Prolegomena of Hypergroup Theory, Second Eddition Aviani, Editor, (1993).
[7] P. Corsini, V. Leoreanu, Applications of Hyperstructure Theory, Advances in Mathematics. Kluwer Aca-
demic Publishers, (2003).
[8] B. Davvaz and V. Leoreanu-Fotea,, Hyperring Theory and Applications, International Academic Press,
Palm Harbor, USA, 2007.
[9] A.J. Diesl, Nil clean rings, J. Algebra, 383 (2013), 197-211.
[10] J. Han and W.K. Nicholson, Extensions of clean rings, Comm. Algebra 29 (2001), No. 6, 2589-2595.
[11] M. Krasner, A Class of Hyperring and Hyper elds, Intern. J. Math. Sci. 6 (1983), 307-312.
[12] C.G. Massouros, On the theory of hyperrings and hyper elds, Algebra i Logika, 24 (1985), 728-742.
[13] F. Marty, Sur une generalization de group, In: 8th Congres Math. Scandinaves: Stockholm, (1934), 45-49.
[14] J. Mittas, Hypergroupes Canoniques, Mathematica Balkanica, 2 (1972), 165-179.
[15] A. Nakassis, Expository and Survey Article Recent Result in hyperring and Hyper eld Theory, Internet. J.
Math and Math. Sci., 11 (2) (1988), 209-220.
[16] W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc., 229 (1977), 278-279.
[17] W. K. Nicholson and Y. Zhou, Clean general rings, J. Algebra, 291 (2005), No. 1, 297-311.
[18] T. Vougiouklis, Hyperstructures and their representations, Hardonic, press, Inc (1994).