# Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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## Bases for the positive cone of a partially ordered moduleHTML articles powered by AMS MathViewer

by W. Russell Belding
Trans. Amer. Math. Soc. 235 (1978), 305-313 Request permission

## 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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