Additive group of a commutative unital ring implies commutator-realizable

Statement

Suppose $G$ is an abelian group that is isomorphic to the additive group of a commutative unital ring. Then, $G$ is a commutator-realizable group. In other words, there exists a group $K$ containing $G$ such that the commutator subgroup of $K$ equals $G$.

Proof

Given: An abelian group $G$ that is the additive group of a commutative unital ring $R$.

To prove: There exists a group $K$ such that the commutator subgroup $[K,K]$ equals $G$.

Proof: Consider the group $U(3,R)$ of upper-triangular $3\times 3$ matrices with $1$s on the diagonal under multiplication. $U(3,R)$ is a group of nilpotency class two, whose commutator subgroup and cneter are both the group of matrices of the form:

${\begin{pmatrix}1&0&a\\0&1&0\\0&0&1\\\end{pmatrix}}$

which is isomorphic to the additive group $G$ of $R$.