# Vector ring

From Encyclopedia of Mathematics

2010 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 $R_\lambda$.

#### References

[1] | L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963) |

**How to Cite This Entry:**

Vector ring.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Vector_ring&oldid=35821

This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article