nLab partially ordered ring




A partially 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, 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 RR is said to be a preordered ring.

Last revised on February 23, 2024 at 19:54:20. See the history of this page for a list of all contributions to it.