Left Self-Distributive Rings and Nearrings

  • Gary F. Birkenmeier


A (near-) ring R is called left self-distributive, LSD, if vxy = vxvy for all v,x,y in R. Right self-distributive (near-) rings, RSD, are defined similarly. A (near-) ring is called self-distributive, SD, if it is both LSD and RSD. Observe that the class of LSD (left near-) rings is exactly the class of (left near-) rings for which each left multiplication mapping (i.e., xax) is a (left near-) ring endomorphism. Hence the class of LSD (left near-)rings includes the AE-(left near-) rings (i.e., those (left near-) rings for which every additive endomorphism is a (left near-) ring endomorphism). In this paper we will discuss the history and recent developments for the class of LSD and related (left near-) rings. Examples will be included to illustrate and delimit the theory.


Commutative Ring Ring Endomorphism Primitive Idempotent Steiner System Semigroup Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J. Angerer and G. Pilz, The structure of near-rings of small order, Lecture Notes in Computer Science No. 144 (Computer Algebra, Marseille 1982), Springer-Verlag, 1982, 57–64.Google Scholar
  2. [2]
    G. M. Bergman, Boolean rings of projection maps, J. London Math. Soc. (2) 4 (1972), 593–598.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    G. F. Birkenmeier, Seminearrings and nearrings induced by the circle operation, Riv. Mat. Pura Appl. 5 (1989), 59–68.MathSciNetMATHGoogle Scholar
  4. [4]
    G. F. Birkenmeier and Y. U. Cho, Additive endomorphisms generated by ring endomorphisms, East-west J. Math. 1 (1998), 73–84.MathSciNetMATHGoogle Scholar
  5. [5]
    G. F. Birkenmeier and H. E. Heatherly, Operation inducing systems, Algebra Universalis 24 (1987), 137–148.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    G. F. Birkenmeier and H. E. Heatherly, Medial near-rings, Monatsh. Math. 107 (1989), 89–110.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    G. F. Birkenmeier and H. E. Heatherly, Polynomial identity properties for near-rings on certain groups, Near-Ring Newsletter 12 (1989), 5–15.Google Scholar
  8. [8]
    G. F. Birkenmeier and H. E. Heatherly, Left self-distributive near-rings, J. Austral. Math. Soc. (Ser. A) 49 (1990), 273–296.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    G. F. Birkenmeier and H. E. Heatherly, Rings whose additive endomorphisms are ring endomorphisms, Bull. Austral. Math. Soc. 42 (1990), 145–152.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    G. F. Birkenmeier and H. E. Heatherly, Self-distributively generated algebras, Contributions to General Algebra 10, Proceedings of the Klagenferrt Conference (eds., D. Dorninger, G. Eigenthaler, H. K. Kaiser, H. Kautschitsch, W. More and W. B. Müller), Verlag Johannes Heyn, Klagenferrt, 1998.Google Scholar
  11. [11]
    G. F. Birkenmeier and H. E. Heatherly, Left self-distributively generated algebras, Comm. Algebra, to appear.Google Scholar
  12. [12]
    G. F. Birkenmeier, H. E. Heatherly, and T. Kepka, Rings with left self distributive multiplication, Acta Math. Hungar. 60 (1992), 107–114.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    M. Ćirić and S. Bogdanović, Rings whose multiplicative semigroups are nil-extensions of a union of groups, Pure Math. Appl. (Ser. A) 1 (1990), 217–234.MATHGoogle Scholar
  14. [14]
    M. Ćirić, S. Bogdanović, and T. Petković, Rings satisfying some semigroup identities, Acta Sci. Math. (Szeged) 61 (1995), 123–137.MathSciNetMATHGoogle Scholar
  15. [15]
    J. R. Clay, The near-rings on groups of low order, Math. Z. 104 (1968), 364–371.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    C. Ferrero Cotti, Sugli stems il cui prodotto è distributivo riśpetto a se stesso, Riv. Mat. Univ. Parma (3) 1 (1972), 203–220.MathSciNetMATHGoogle Scholar
  17. [17]
    P. Dehornoy, An alternative proof of Laver’s results on the algebra generated by an elementary embedding, Set Theory of the Continuum (Eds. H. Judah, W. Just, H. Woodin), Springer-Verlag, New York, 1992, pp. 27–33.CrossRefGoogle Scholar
  18. [18]
    S. Dhompongsa and J. Sanwong, Rings in which additive mappings are multiplicative, Studia Sci. Math. Hungar. 22 (1987), 357–359.MathSciNetMATHGoogle Scholar
  19. [19]
    A. Drápai, Finite left distributive algebras with one generator, J. Pure Appl. Algebra 121 (1997), 233–251.MathSciNetCrossRefGoogle Scholar
  20. [20]
    M. Dugas, J. Hausen, and J. A. Johnson, Rings whose additive endomorphisms are multiplicative, Period. Math. Hungar. 23 (1991), 65–73.MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    M. Dugas, J. Hausen, J. A. Johnson, Rings whose additive endomorphisms are ring endomorphisms, Bull. Austral. Math. Soc. 45 (1992), 91–103.MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    S. Feigelstock, Rings whose additive endomorphisms are multiplicative. Period. Math. Hungar. 19 (1988), 257–260.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    S. Feigelstock, Rings whose additive endomorphisms are N-multiplicative, Bull. Austral. Math. Soc. 39 (1989), 11–14.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    S. Feigelstock, Rings whose additive endomorphisms are n-multiplicative II, Period. Math. Hungar. 25 (1992), 21–26.MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    S. Feigelstock and R. Raphael, Distributive multiplication rings, Period. Math. Hungar. 25 (1992), 161–165.MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    B. Ganter and H. Werner, Co-ordinatizing Steiner systems, Topics on Steiner Systems (Eds. C. C. Linder and A. Rosa), North-Holland, Amsterdam, 1980, pp. 3–24.CrossRefGoogle Scholar
  27. [27]
    R. Hill, The additive group of commutative rings generated by idempotents, Proc. Amer. Math. Soc. 38 (1973), 499–502.MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    Y. Hirano, On rings whose additive endomorphisms are multiplicative, Period. Math. Hungar. 23 (1991), 87–89.MathSciNetMATHCrossRefGoogle Scholar
  29. [29]
    J. Ježek and T. Kepka, Distributive groupoids and symmetry-by-mediality, Algebra Universalis 19 (1984), 208–216.MathSciNetMATHCrossRefGoogle Scholar
  30. [30]
    J. Ježek, T. Kepka, and P. Němec, Distributive Groupoids, Rozpravy Ceskoslovenske Acad. Ved. Rada Mat. Prirod. Ved., 91/3, Prague, 1981.Google Scholar
  31. [31]
    D. Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23 (1982), 37–66.MathSciNetMATHCrossRefGoogle Scholar
  32. [32]
    A. V. Kelarev, On left self distributive rings, Acta Math. Hungar. 71 (1996), 121–122.MathSciNetMATHCrossRefGoogle Scholar
  33. [33]
    T. Kepka, Varieties of left distributive semigroups, Acta Univ. Carolin. Math. Phys. 25 (1984), 3–18.MathSciNetMATHGoogle Scholar
  34. [34]
    T. Kepka, Semirings whose additive endomorphisms are multiplicative, Comment. Math. Univ. Carolinae 34 (1993), 213–219.MathSciNetMATHGoogle Scholar
  35. [35]
    T. Kepka and P. Němec, Distributive groupoids and the finite basis property, J. Algebra 70 (1981), 229–237.MathSciNetMATHCrossRefGoogle Scholar
  36. [36]
    K. H. Kim and F. W. Roush, Additive endomorphisms of rings, Period. Math. Hungar. 12 (1981), 241–242.MathSciNetMATHCrossRefGoogle Scholar
  37. [37]
    J. J. Malone, Near-rings with trivial multiplication, Amer. Math. Monthly 74 (1967), 1111–1112.MathSciNetMATHCrossRefGoogle Scholar
  38. [38]
    J. J. Malone, D. G. near-rings on the infinite dihedral group, Proc. Royal Soc. Edinburgh 78A (1977), 67–70.MathSciNetGoogle Scholar
  39. [39]
    S. V. Matveev, Distributive groupoids in knot theory, Math. USSR Sbornik. 47 (1984), 73–83.CrossRefGoogle Scholar
  40. [40]
    C. J. Maxson, Idempotent generated algebras and Boolean pairs, Fund. Math. 93 (1976), 15–22.MathSciNetMATHGoogle Scholar
  41. [41]
    Lee Sin-Min, Lattice ofequational subclasses of distributive semigroups, Nanta Math. J. 9 (1976), 65–69.MATHGoogle Scholar
  42. [42]
    M. Petrich, Structure des demi-groupes et anneaux distributifs, C. R. Acad. Sc. Paris Sec. A-B 268 (1969), A849–A852.MathSciNetGoogle Scholar
  43. [43]
    G. Pilz, List of low order near-rings, along with some special properties, Near-Ring Newsletter 2 (1980), 5–23.Google Scholar
  44. [44]
    G. Pilz, Near-rings, North-Holland, Amsterdam, 1983.MATHGoogle Scholar
  45. [45]
    A. Romanowska and B. Roszkowska, Algebra Universalis 26 (1989), 7–15.MathSciNetMATHCrossRefGoogle Scholar
  46. [46]
    R. Scapellato, On autodistributive near-rings, Riv. Mat. Univ. Parma 10 (1984), 303–310.MathSciNetMATHGoogle Scholar
  47. [47]
    R. P. Sullivan, Research problem No. 23, Period. Math. Hungar. 8 (1977), 313–314.MathSciNetMATHCrossRefGoogle Scholar
  48. [48]
    M. Willhite, Distributively generated near-rings on the dihedral group of order eight, M. S. Thesis, Texas A&M Univ., College Station, 1970.Google Scholar
  49. [49]
    R. Yearbly and H. Heatherly, The near-ring multiplications on Alt(4), Near-Ring Newsletter 6 (1983), 59–73.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Gary F. Birkenmeier
    • 1
  1. 1.Department of MathematicsUniversity of Southwestern LouisianaLafayetteUSA

Personalised recommendations