nLab partially ordered ring

Definition

A partially ordered ring is a ring $R$ with a partial order such that for all elements $a,b,c$ in $R$, $a \leq b$ implies $a + c \leq b + c$, and $0 \leq a$ and $0 \leq b$ implies $0 \leq a \cdot b$.

Due to the reflexivity of the partial order, partially ordered rings may have zero divisors. Also, the trivial ring is an partially ordered ring.

If the relation $\leq$ is only a preorder, then the ring $R$ is said to be a preordered ring.

Related concepts

• lattice-ordered ring

• totally ordered ring

• pseudo-ordered rings

• ordered field

• ordered integral domain

• ordered group

• protoring

External links

• Wikipedia, Partially ordered ring