Subgroups of finite index in an additive group of a ring

Abstract

If K is an infinite field and G⫅K is a subgroup of finite index in an additive group, then K∗=G∗G∗−1 where G∗ denotes the set of all invertible elements in G and G∗−1 denotes all inverses of elements of G∗. Similar results hold for various fields, division rings and rings.