Left Euclidean (2, 2)-Algebras

The concept of a commutative and zero-divisor-free Euclidean ring, defined via an Euclidean function, has been generalized to arbitrary left Euclidean rings and than to various other structures as semirings, nearrings and semi-near-rings. As first shown in the dissertation (Hebisch, 1984 ), these di...

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Bibliographic Details
Published in:Communications in algebra 2007-06, Vol.35 (6), p.2035-2055
Main Authors: Hebisch, Udo, Weinert, Hanns Joachim
Format: Article
Language:English
Subjects:
Ascending chain condition
Euclidean algorithm
Left Euclidean functions
Left Euclidean rings
Primary 16-99
Secondary 16Y30, 16Y60, 16U30
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Summary:The concept of a commutative and zero-divisor-free Euclidean ring, defined via an Euclidean function, has been generalized to arbitrary left Euclidean rings and than to various other structures as semirings, nearrings and semi-near-rings. As first shown in the dissertation (Hebisch, 1984 ), these different investigations can be combined considering arbitrary (2, 2)-algebras (S, +, ·), defined as left Euclidean in a suitable way. Here we present and investigate an improved version of this concept. Moreover, Motzkin ( 1949 ) gave a criterion which characterizes a commutative and zero-divisor-free ring as Euclidean by certain chains of product ideals, without the use of Euclidean functions. In the central part of this paper we obtain a corresponding characterization and two further criterions, necessary and sufficient for an algebra (S, +, ·) to be left Euclidean. Based on this we prove several results on these algebras.
ISSN:0092-7872
1532-4125
DOI:10.1080/00927870601139534