For each a, b R let a b = a+ b + ab. (a)

For each a, b ϵ R let a º b = a+ b + ab. 

(a) ° 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 °.

(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.