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).