A Property of the Commutator Ideal of a Finitely Generated Metabelian D-group

A D-group is a group G with the property that, for every element g in G and every positive integer n,g has a unique nth root in G. G is termed metabelian if its commutator subgroup ] , [ G G is abelian. Not every subgroup of a D-group is also a D-group. If a subgroup is a D-group we call the subgroup “sub- D -group.” The “commutator ideal” of G is the smallest sub-D-group containing its commutator subgroup. We found that the commutator ideal can be viewed as a finitely generated module over a ring when G is a finitely generated metabelian D-group.