# partially ordered ring

A ring $R$ that is a poset at the same time is called a partially ordered ring, or a po-ring, if, for $a,b,c\in R$,

• $a\leq b$ implies $a+c\leq b+c$, and

• $0\leq a$ and $0\leq b$ implies $0\leq ab$.

Note that $R$ does not have to be associative.

If the underlying poset of a po-ring $R$ is in fact a lattice, then $R$ is called a lattice-ordered ring, or an l-ring for short.

Remark. The underlying abelian group of a po-ring (with addition being the binary operation) is a po-group. The same is true for l-rings.

Below are some examples of po-rings:

• Clearly, any (totally) ordered ring is a po-ring.

• Any matrix ring over an ordered field is an l-ring if we define $(a_{ij})\leq(b_{ij})$ whenever $a_{ij}\leq b_{ij}$ for all $i,j$.

Remark. Let $R$ be a po-ring. The set $R^{+}:=\{r\in R\mid 0\leq r\}$ is called the positive cone of $R$.

## References

 Title partially ordered ring Canonical name PartiallyOrderedRing Date of creation 2013-03-22 16:55:04 Last modified on 2013-03-22 16:55:04 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 8 Author CWoo (3771) Entry type Definition Classification msc 13J25 Classification msc 16W80 Classification msc 06F25 Synonym po-ring Synonym l-ring Synonym lattice-ordered ring Defines lattice ordered ring Defines positive cone