nLab
ordered ring

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Definition

An ordered ring is a ring RR with a partial order such that for all elements a,b,ca,b,c in RR, aba \leq b implies a+cb+ca + c \leq b + c, and 0a0 \leq a and 0b0 \leq b implies 0ab0 \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.