   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
DOI: https://doi.org/10.1090/S0002-9947-1978-0472640-8
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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