A ring R over a commutative ring K, that has a basis of elements g1, g2
, … , gn
forming a group G under multiplication, is called a group ring of G over K. Since all group rings of a given G over a given K are isomorphic, we may speak of the group ring KG of G over X.
Let π be any partition of G into non-empty sets GA, GB
, … . Any subring P of KG that has a basis of elements
is a partition ring of G over K.
If P is a partition ring of G over Z, the ring of integers, then the basis A, B, … for P clearly serves as a basis for a partition ring P’ = Q ⊗ P of G over Q, the field of rationals.