R is a division ring if and only if every element of R except one is left quasi-regular.

Data from exercise 1

 

For each a, b ϵ R let a º b = a+ b + ab. (a) o is an associative binary operation with identity element O ϵ R.

(b) The set G of all elements of R that are both left and right quasi-regular forms a group under º.

(c) If R has an identity, then a ϵ R is left [resp. right] quasi-regular if and only if 1_R+ a is left [resp. right] invertible.