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A Characterization of Left Regularity

Year 2019, Volume: 2 Issue: 1, 11 - 13, 20.03.2019
https://doi.org/10.32323/ujma.469745

Abstract

We show that a zero-symmetric near-ring $N$ is left regular if and only if $N $ is regular and isomorphic to a subdirect product of integral near-rings, where each component is either an Anshel-Clay near-ring or a trivial integral near-ring. We also show that a zero-symmetric near-ring is regular without nonzero nilpotent elements if and only if the multiplicative semigroup of N is a union of disjoint groups.

References

  • [1] G. Pilz, Near-rings, The theory and its applications, 2 ed., North-Holland Mathematics Studies 23, North-Holland Publishing Co., Amsterdam, 1983, MR0721171.
  • [2] J. Andre, H. H. Ney, On Anshel-Clay-nearrings, Near-rings and near-fields (Oberwolfach, 1989), 15-20, Math. Forschungsinst. Oberwolfach, Schwarzwald, 1995. MR1725312
  • [3] M. Anshel, J. R. Clay, Planar algebraic systems: Some geometric interpretations, J. Algebra 10 (1968), 166-173, MR0241473.
  • [4] G. Ferrero, Stems planari e BIB-disegni, Riv. Mat. Univ. Parma, 2(11) (1970), 79-96, MR031332
  • [5] G. Szeto, The Sub-Semi-Groups excluding zero of a near-ring, Monatsh. Math., 77 (1973), 357-362, MR0330237.
  • [6] G. Szeto, Planar and strongly uniform near-rings, Proc. Amer. Math. Soc., 44 (1974), 269-274, MR0340351.
  • [7] H. H. Ney, Anshel-Clay near-rings and semiaffine parallelogramspaces, Near-rings and near-fields (Fredericton, NB, 1993), 203-207, Math. Appl., 336, Kluwer Acad. Publ., Dordrecht, 1995, MR1366480.
  • [8] G. Mason, Strongly regular near-rings, Proc. Edinburgh Math. Soc., 23(1) (1980), 27-35, MR0582019.
  • [9] A. H. Clifford, G. B. Preston, The algebraic theory of semigroups, 1(7), Mathematical Surveys, Amer. Math. Soc., Providence, R.I., (1977), MR0132791 7.
  • [10] Y. U. Cho, A study on near-rings with semi-central idempotents, ar East J. Math. Sci., 98(6), (2015), 759-762.
  • [11] H. E. Bell, Near-rings in which each element is a power of itself, Bull. Austral. Math. Soc., 2 (1970), 363-368, MR0263877.
  • [12] V. G. Marin, Some properties of regular near-algebras, Ring-theoretical constructions (Russian), Mat. Issled, 49 (1979), 105-114, 162-163, MR0544982.
Year 2019, Volume: 2 Issue: 1, 11 - 13, 20.03.2019
https://doi.org/10.32323/ujma.469745

Abstract

References

  • [1] G. Pilz, Near-rings, The theory and its applications, 2 ed., North-Holland Mathematics Studies 23, North-Holland Publishing Co., Amsterdam, 1983, MR0721171.
  • [2] J. Andre, H. H. Ney, On Anshel-Clay-nearrings, Near-rings and near-fields (Oberwolfach, 1989), 15-20, Math. Forschungsinst. Oberwolfach, Schwarzwald, 1995. MR1725312
  • [3] M. Anshel, J. R. Clay, Planar algebraic systems: Some geometric interpretations, J. Algebra 10 (1968), 166-173, MR0241473.
  • [4] G. Ferrero, Stems planari e BIB-disegni, Riv. Mat. Univ. Parma, 2(11) (1970), 79-96, MR031332
  • [5] G. Szeto, The Sub-Semi-Groups excluding zero of a near-ring, Monatsh. Math., 77 (1973), 357-362, MR0330237.
  • [6] G. Szeto, Planar and strongly uniform near-rings, Proc. Amer. Math. Soc., 44 (1974), 269-274, MR0340351.
  • [7] H. H. Ney, Anshel-Clay near-rings and semiaffine parallelogramspaces, Near-rings and near-fields (Fredericton, NB, 1993), 203-207, Math. Appl., 336, Kluwer Acad. Publ., Dordrecht, 1995, MR1366480.
  • [8] G. Mason, Strongly regular near-rings, Proc. Edinburgh Math. Soc., 23(1) (1980), 27-35, MR0582019.
  • [9] A. H. Clifford, G. B. Preston, The algebraic theory of semigroups, 1(7), Mathematical Surveys, Amer. Math. Soc., Providence, R.I., (1977), MR0132791 7.
  • [10] Y. U. Cho, A study on near-rings with semi-central idempotents, ar East J. Math. Sci., 98(6), (2015), 759-762.
  • [11] H. E. Bell, Near-rings in which each element is a power of itself, Bull. Austral. Math. Soc., 2 (1970), 363-368, MR0263877.
  • [12] V. G. Marin, Some properties of regular near-algebras, Ring-theoretical constructions (Russian), Mat. Issled, 49 (1979), 105-114, 162-163, MR0544982.
There are 12 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Peter Fuchs 0000-0001-9165-3688

Publication Date March 20, 2019
Submission Date October 12, 2018
Acceptance Date December 6, 2018
Published in Issue Year 2019 Volume: 2 Issue: 1

Cite

APA Fuchs, P. (2019). A Characterization of Left Regularity. Universal Journal of Mathematics and Applications, 2(1), 11-13. https://doi.org/10.32323/ujma.469745
AMA Fuchs P. A Characterization of Left Regularity. Univ. J. Math. Appl. March 2019;2(1):11-13. doi:10.32323/ujma.469745
Chicago Fuchs, Peter. “A Characterization of Left Regularity”. Universal Journal of Mathematics and Applications 2, no. 1 (March 2019): 11-13. https://doi.org/10.32323/ujma.469745.
EndNote Fuchs P (March 1, 2019) A Characterization of Left Regularity. Universal Journal of Mathematics and Applications 2 1 11–13.
IEEE P. Fuchs, “A Characterization of Left Regularity”, Univ. J. Math. Appl., vol. 2, no. 1, pp. 11–13, 2019, doi: 10.32323/ujma.469745.
ISNAD Fuchs, Peter. “A Characterization of Left Regularity”. Universal Journal of Mathematics and Applications 2/1 (March 2019), 11-13. https://doi.org/10.32323/ujma.469745.
JAMA Fuchs P. A Characterization of Left Regularity. Univ. J. Math. Appl. 2019;2:11–13.
MLA Fuchs, Peter. “A Characterization of Left Regularity”. Universal Journal of Mathematics and Applications, vol. 2, no. 1, 2019, pp. 11-13, doi:10.32323/ujma.469745.
Vancouver Fuchs P. A Characterization of Left Regularity. Univ. J. Math. Appl. 2019;2(1):11-3.

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