Left Self-Distributive Rings and Nearrings
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., x → ax) 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.
Unable to display preview. Download preview PDF.
- 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
- 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
- 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
- G. F. Birkenmeier and H. E. Heatherly, Left self-distributively generated algebras, Comm. Algebra, to appear.Google Scholar
- J. Ježek, T. Kepka, and P. Němec, Distributive Groupoids, Rozpravy Ceskoslovenske Acad. Ved. Rada Mat. Prirod. Ved., 91/3, Prague, 1981.Google Scholar
- G. Pilz, List of low order near-rings, along with some special properties, Near-Ring Newsletter 2 (1980), 5–23.Google Scholar
- 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
- R. Yearbly and H. Heatherly, The near-ring multiplications on Alt(4), Near-Ring Newsletter 6 (1983), 59–73.Google Scholar