Let $M$ be a proper $\Bbb Z$-submodule of $\Bbb Q.$ Can we say that $M$ is finitely generated?
I know that $\Bbb Q$ is not finitely generated as a $\Bbb Z$-module.
Please help me in this regard. Thank you very much.
2026-03-26 17:13:18.1774545198
Proper $\Bbb Z$-submodules of $\Bbb Q$ are finitely generated or not?
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Take a finite set of rational numbers, and express them in simplest form. The denominators of these numbers when factorized will invovle a finite number of primes.
FACT 1: If a set of rational numbers involve only a finite number of primes this way, them their sum and their products will also involve THE SAME set of primes numbers or a subset, but not a larger set. (Look at LCM)
As there are infinitely many prime numbers we can take a proper infinite subset of those prime numbers and the collection of rational numbers such that their denominators are factorizable using only those prime numbers. This submodule cannot be finitely generated and is proper.
Lord Shark's short comment mentions this fact taking the subset to be all odd primes (omitting 2, so a proper subset).