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Left Self-Distributive Rings and Nearrings

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Near-Rings and Near-Fields

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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.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Southwestern Louisiana, Lafayette, LA, 70504, USA

    Gary F. Birkenmeier

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  1. Gary F. Birkenmeier
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Editor information

Editors and Affiliations

  1. Department of Mathematics, National Cheng Kung University, Tainan, Taiwan, Republic of China

    Yuen Fong

  2. Department of Mathematics, Texas A & M University, College Station, Texas, USA

    Carl Maxson

  3. Department of Mathematics, University of Edinburgh, Edinburgh, Scotland

    John Meldrum

  4. Institute for Mathematics, Johannes Kepler Universität Linz, Linz, Austria

    Günter Pilz

  5. Department of Mathematics, University of Stellenbosch, Stellenbosch, South Africa

    Andries van der Walt  & Leon van Wyk  & 

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© 2001 Springer Science+Business Media Dordrecht

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Birkenmeier, G.F. (2001). Left Self-Distributive Rings and Nearrings. In: Fong, Y., Maxson, C., Meldrum, J., Pilz, G., van der Walt, A., van Wyk, L. (eds) Near-Rings and Near-Fields. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0954-6_2

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  • DOI: https://doi.org/10.1007/978-94-010-0954-6_2

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-3802-7

  • Online ISBN: 978-94-010-0954-6

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