Computing Quotient Groups with Infinite Groups

I've asked a similar question: Computing Quotient Groups

But now I want to compute a quotient group involving a direct product in which every direct factor is infinite. For example $\mathbb{Z} \times \mathbb{Z} \times \mathbb{Z} / \langle(1, 1, 1)\rangle$ or $\mathbb{Z} \times \mathbb{Z} \times \mathbb{Z} / \langle(3, 3, 3)\rangle$. Is there a better approach than just looking for a homomorphism?

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"The direct product of all infinite groups" would be a very big thing indeed. I think you mean "a direct product in which every direct factor is infinite." –  Arturo Magidin Dec 13 '11 at 6:00
@Arturo: Thanks for the suggestion. –  Student Dec 13 '11 at 6:02

In the case of a finite direct product of copies of $\mathbb{Z}$, the Smith Normal Form solves the problem for you.
Here, note that $\mathbb{Z}^3$ has a basis of the form $(1,0,0)$, $(1,1,0)$, and $(1,1,1)$, so your first quotient is just isomorphic to $\mathbb{Z}^2$ (you are just "killing" one generator). The same basis tells you that the second quotient is isomorphic to $\mathbb{Z}^2\times \mathbb{Z}_3$.