Vector ring

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2020 Mathematics Subject Classification: Primary: 06F25 [MSN][ZBL]

A partially ordered ring $R$ (cf. Partially ordered set, Ordered ring) which is expressible as a subdirect sum of totally ordered rings $R_\lambda$ (cf. Totally ordered set). Each element of a vector ring is a vector $\mathbf{a} = (\ldots,a_\lambda,\ldots)$with coordinates in $R_\lambda$, and $\mathbf{a} \ge 0$ if and only if each $a_\lambda \ge 0$.

If the partial order of $R$ is an intersection of total orders, then $R$ is a vector ring, and $R$ itself, provided with various linear extensions of its partial order, may be taken as $R_\lambda$.


[1] L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963)
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Vector ring. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article