I'm teaching an Advanced Algebra seminar and presented a proof of the Fundamental Theorem of Abelian Groups that followed the proof presented by Herstein in Topics in Algebra. In that proof you look at relations among generators of the group G and use that to peel away cyclic groups as direct summands of G, which leave the infinite sums at the end. In the proof you treat the finite and the infinite summands slightly differently.
A student in the class modified the proof by ordering the integers, not by their size, but by the divisibility relation $a\leq b$ iff $a\vert b$. If we define $\mathbb{Z}_0= \mathbb{Z}/0\mathbb{Z} =\mathbb{Z}$, one can treat the integer summands in the same manner as the finite summands. The $\mathbb{Z}^r$ summands still show up at the end of the proof, but now correspond to $0$ being the "only element" left in the set of coefficients of the equations relating the generators.
My question is:
Is a proof along these lines known?
I haven't found it in my review of the proofs available online or in my algebra texts. If this idea is already known, can someone publish a reference? Thank you!