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).