Unit neighborhoods of zero in topological ordered rings

Abstract

A closed unit neighborhood of zero in a topological ring is an additively symmetric and multiplicatively idempotent regular closed neighborhood of zero containing the unity whose interior is multiplicatively idempotent as well. The search for nontrivial closed unit neighborhoods of zero in topological rings is an ongoing quest. A unital ordered ring is a ring endowed with a partial ordering compatible with the addition and multiplication by positive elements for which zero and the unity are comparable. A topological ordered ring is a unital ordered ring for which the order topology is a ring topology. Recently, it was posed the question whether the set of elements lying in between −1 and 1 is a closed unit neighborhood of 0 in a topological ordered ring. This question has been partially solved on topological totally ordered division rings with no holes. Here, we provide a full answer in topological ordered rings (not relying on total orderings nor on division rings).