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, 0 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).