Group Ring

• The ring defined as follows is called the group ring of over .

  • consists of the set of all formal sums , where and where all but a finite number of are
  • The sum of two elements of is defined as
  • The product of two multiplications is defined using multiplications for both and as follows
    Intuitively, we distribute the sums to get and replace that with .

• If is a field, we call as the group algebra of over .

• (Fraleigh 24.4) If is any group written multiplicatively and is a Commutative Ring with nonzero unity then is a ring.

Links §

• Fraleigh