Left Euclidean (2, 2)-Algebras

Abstract

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.