Linear orders on rings

Szigeti, Jenő (2005) Linear orders on rings. COMMUNICATIONS IN ALGEBRA, 33 (8). pp. 2683-2695. ISSN 0092-7872

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Abstract

The general ideas introduced in [8] are adopted in order to investigate the quasi cones and the cones of a ring. Using the finite extension property for cones, we answer the question, when a partial order in a partially ordered ring has a compatible linear extension (equivalently, when the positive cone is contained in a full cone). It turns out, that if there is no such extension, then it is caused by a finite system of polynomial like equations satisfied by some elements of a certain finite subset of the ring and some positive elements. As a consequence, we prove that a partial order can be linearly extended if and only if it can be linearly extended on every finitely generated subring.

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