Left Self-Distributive Rings and Nearrings

Gary F. Birkenmeier
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Gary F. Birkenmeier: University of Southwestern Louisiana, Department of Mathematics

A chapter in Near-Rings and Near-Fields, 2001, pp 10-22 from Springer

Abstract: Abstract 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.

Keywords: Commutative Ring; Ring Endomorphism; Primitive Idempotent; Steiner System; Semigroup Algebra (search for similar items in EconPapers)
Date: 2001
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DOI: 10.1007/978-94-010-0954-6_2

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