Free abelian group on prime divisors.

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Can someone tell about how it is possible to write locally finite sets $n_{Z}$ here with the free abelian group generated by the prime divisors?

First I thought it was a cartesian product of spaces but it is in a free abelian group, so I dont know how to continue.

It looks like I use the prime divisors to generate the locally finite subsets, is that all I have to do ?