On the biordered set of rings

Abstract

In [4] K.S.S. Nambooripad introduced biordered sets as a partial algebra $\left(E, \omega^r, \omega^l\right)$ where $\omega^r$ and $\omega^l$ are two quasiorders on the set $E$ satisfying biorder axioms; to study the structure of a regular semigroup. Later in [2] David Esdown showed that the set of idempotents of a regular semigroup forms a regular biordered set. Here we extend the idea of biordered sets into rings and discussed some of its properties.