nLab
ordered ring

Contents

Contents

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.

Last revised on June 18, 2021 at 17:40:24. See the history of this page for a list of all contributions to it.