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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Bases for the positive cone of a partially ordered module


Author: W. Russell Belding
Journal: Trans. Amer. Math. Soc. 235 (1978), 305-313
MSC: Primary 06A75
MathSciNet review: 0472640
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Abstract: $ (R,{R^ + })$ is a partially ordered ring and $ (M,{M^ + })$ is a strict $ (R,{R^ + })$-module. So M is a left R-module and $ ({R^ + }\backslash \{ 0\} )({M^ + }\backslash \{ 0\} ) \subseteq {M^ + }\backslash \{ 0\} $. Let $ \leqslant '$ be the order induced on M by $ {M^ + }.B \subseteq {M^ + }$ is an $ {R^ + }$-basis for $ {M^ + }$ means $ {R^ + }B = {M^ + }$ (spanning) and if r is in R, b in B with $ 0 < 'rb \leqslant 'b$ then $ rb \notin {R^ + }(B\backslash \{ b\} )$ (independence).

Result: If B and D are $ {R^ + }$-bases for $ {M^ + }$ then card $ B = $ card D and to within a permutation $ {b_i} = {u_i}{d_i}$, for units $ {u_i}$ of $ {R^ + }$.


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DOI: https://doi.org/10.1090/S0002-9947-1978-0472640-8
Keywords: Partially ordered ring, partially ordered module, directed, order preserving automorphism group, closure operator
Article copyright: © Copyright 1978 American Mathematical Society