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A partially ordered ring $R$ (cf. [[Partially ordered set]]) 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 $R_\lambda$.

====References====
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<TR><TD valign="top">[1]</TD> <TD valign="top">  L. Fuchs,   "Partially ordered algebraic systems" , Pergamon  (1963)</TD></TR>
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