I'm trying to solve a question which asks me to determine whether the quotient $\mathbb{Z}$-module $M=\frac{\mathbb{Z}^3}{\langle (3,3,1),(2,2,2)\rangle}$ is free. I'm then supposed to find some element of $M$ of infinite order.

I'm really new to modules and finding them hard to wrap my head around, so would appreciate any help you could offer/hints you could give me.

EDIT: while a similar question has been asked here - Show that $M=\frac{\mathbb{Z}^3}{\langle (3,3,1),(2,2,2)\rangle}$ is isomorphic to $\mathbb{Z} \oplus \mathbb{Z}_4$ as $\mathbb{Z}$-modules - the answers there use the fact $M$ is isomorphic to $\mathbb{Z} \oplus \mathbb{Z}_4$. I'd like to show that it isn't free more directly if at all possible. Would appreciate any help anyone could offer.


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