一個群G 其每一個元素都有唯一的n 次方根，n 為任意正整數，則G 稱為可除群(D-group)。若其導出子群[G ,G] 為可交換的，則稱G為亞交換群。並非每個子群都是一個可除群，若一個子群是可除群，我們稱之為子可除群(sub-D-group)，而G 的交換子理想(commutator ideal)則是在G中包含其導出群的最小子可除群，我們發現當G是一個有限生成亞交換可除群時，其交換子理想是一個有限生成模。
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.
