An 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, ordered rings may have zero divisors. Also, the trivial ring is an ordered ring.
Wikipedia, Partially ordered ring
Wikipedia, Ordered ring