If R is a ring (this includes the important case where R is a field), it has a simple additive group structure: just take all its elements with the addition operator. This gives an abelian group, and is boring.

We'd like to do the same with multiplication. Of course, not all of R's elements are invertible (for instance, never is!), so we define R* as the set of all invertible elements r of R (i.e. for which there exists s in R such that r s = s r = 1), equipped with the multiplication operation. This is a group, and tends to be interesting. If R is commutative, R* is abelian (this includes the case when R is a field).