Hostname: page-component-84b7d79bbc-g5fl4 Total loading time: 0 Render date: 2024-07-29T09:16:28.570Z Has data issue: false hasContentIssue false
  • English
  • Français

Left self distributive near-rings

Published online by Cambridge University Press:  09 April 2009

Gary Birenmeier  and
Henry Heatherly
Gary Birenmeier
Affiliation:
University of Southwestern LouisianaLafayette, Louisiana 70504, U.S.A.
Henry Heatherly
Affiliation:
University of Southwestern LouisianaLafayette, Louisiana 70504, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper considers (left) near-rings which satisfy the left self distributive (LSD) identity: abc = abac. This is exactly the class of near-rings for which each left multiplication mapping, τa: x → ax, is a near-ring endomorphism. Simple and subdirectly irreducible ones are classified and semidirect sum decompositions into reduced and nilpotent pieces are given. LSD near-rings with restrictive conditions on nilpotent elements or annihilating sets are considered. Type 1 prime (semiprime) ideals in an LSD near-ring are completely prime (semiprime). Further results on prime and maximal ideals are given. Numerous examples are given to illuminate the theory and to illustrate its limitations. Some analogous theory for right self distributive near-rings is given (those satisfying the identity: abc = acbc).

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 49 , Issue 2 , October 1990 , pp. 273 - 296
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]André, J., ‘Non-commutative geometry, near-rings and near-fields,’ Proc. Conf. Near-rings and Near-fields, edited by Betsch, G., pp. 1–14 (Amsterdam, 1987).Google Scholar
[2]Angerer, J. and Pilz, G., ‘The structure of near-rings of small order,’ Lecture Notes in Computer Science No. 144, Computer Algebra, Marseille 1982, pp. 57–64, Springer-Verlag, 1982.CrossRefGoogle Scholar
[3]Birkenmeier, G. and Heatherly, H., ‘Operation inducing systems,’ Algebra Universalis 24 (1987), 137–148.CrossRefGoogle Scholar
[4]Birkenmeier, G. and Heatherly, H., ‘Medial near-rings,’ Monatsh. Math., 107 (1989), 89–110.Google Scholar
[5]Birkenmeier, G. and Heatherly, H., ‘Medial rings and associated radical,’ submitted.Google Scholar
[6]Birkenmeier, G. and Heatherly, H., ‘Medial near-rings: prime ideals and reduced near-rings,’ submitted.Google Scholar
[7]Birkenmeier, G., Heatherly, H. and Kepka, T., ‘Rings with left self distributive multiplication,’ submitted.Google Scholar
[8]Birkhoff, G., ‘Subdirect unions in universal algebra,’ Bull. Amer. Math. Soc. 50 (1944), 764–768.Google Scholar
[9]Clay, J. R., ‘The near-rings on groups of low order,’ Math. Z. 104 (1968), 364–371.CrossRefGoogle Scholar
[10]Clay, J. R., ‘Suggested direction for future research in near-ring theory,’ Proc. Conf Near-rings and Near-fields, edited by Ferrero, G. and Cotti, C. Ferrero, pp. 13–24 (San Benedetto del Tronto, 1981).Google Scholar
[11]Dhompongsa, S. and Sanwong, J., ‘Rings in which additive mappings are multiplicative,’ Studia Sci. Math. Hungar. 22 (1987), 357–359.Google Scholar
[12]Cotti, C. Ferrero, ‘Sugli stems il cui prodotto è distributivo rispetto a se stesso,’ Riv. Mat. Univ. Parma (3) 1 (1972), 203–220.Google Scholar
[13]Gonshor, H., ‘On abstract affine near-rings,’, Pacific. J. Math. 14 (1964), 1237–1240.CrossRefGoogle Scholar
[14]Kepka, T., ‘Varieties of left distributive semigroups,’ Acta Univ. Carolinae-Math. et Phys. 25 (1984), 3–18.Google Scholar
[15]Ligh, S., ‘The structure of a special class of near-rings,’ J. Austral. Math. Soc. Ser. A 13 (1972), 141–146.CrossRefGoogle Scholar
[16]Malone, J. J., ‘Near-rings with trivial multiplication,’ Amer. Math. Monthly 74 (1967), 1111–1112.Google Scholar
[17]Malone, J. J., ‘D.G. near-rings on the infinite dihedral group,’ Proc. Royal Soc. Edinburgh 78A (1977), 67–70.Google Scholar
[18]Malone, J. J., ‘D. G. near-rings on D2n, n even,’ Worcester Polytechnic Inst. Math. Rep., 1978.Google Scholar
[19]Meldrum, J. D. P., Near-rings and their links with groups, (Pitman Research Notes No. 134, London, 1985).Google Scholar
[20]Petrich, M., ‘Structure des demi-groupes et anneaux distributifs,’ C. R. Acad. Sci. Paris Sér. A-B 268 (1969), A849–A852.Google Scholar
[21]Pilz, G., ‘List of low order near-rings, along with some special properties,’ Near-Ring Newsletter 2 (1980), 5–23.Google Scholar
[22]Pilz, G., Near-rings. (North-Holland, Amsterdam, 1983).Google Scholar
[23]Ramakotaiah, D. and Rao, G. K., ‘IFP near-rings,’ J. Austrah Math. Soc. Ser. A 27 (1979), 365–370.CrossRefGoogle Scholar
[24]Scapellato, R., ‘On geometric near-rings,’ Boll. Un. Mat. Ital. A (6) 2 (1983), 389–393.Google Scholar
[25]Scapellato, R., ‘On autodistributive near-rings,’ Riv. Mat. Univ. Parma (4) 10 (1984), 303–310.Google Scholar
[26]Scapellato, R., ‘On geometric near-rings,’ Proc. Conf. Near-rings and Near-fields, edited by Betsch, G., pp. 253–254 (Amsterdam, 1987).Google Scholar
[27]Scapellato, R., ‘Strongly monogenic near-rings,’ Arch. Math. 51 (1988), 476–480.CrossRefGoogle Scholar
[28]Subramanyan, N. V., ‘Boolean semigroups,’ Math. Ann. 148 (1962), 395–401.CrossRefGoogle Scholar
[29]Willhite, M., Distributively generated near-rings on the dihedral group of order eight, (M. S. Thesis, Texas A&M Univ., College Station, 1970).Google Scholar
[30]Yearby, R. and Heatherly, H., ‘The near-ring multiplications on Alt(4),’ Near-Ring Newsletter 6 (1983), 59–73.Google Scholar