# Vector ring

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 $R_\lambda$.