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I'm working on a music-theoretical problem involving products and ratios of smooth numbers. Currently I'm studying 5-smooth numbers but I'm planning to examine higher primes too. I'm not an expert of abstract algebra, Wikipedia is my guide so far. I believe that understanding the algebraic aspects of my subject would lead to deeper results.

I assume that the set $S_n$ of all $n$-smooth numbers forms an abelian group under multiplication (addition is irrelevant to my application). What is the related algebraic structure for fractions of $n$-smooth numbers?

Field of fractions doesn't seem to work here as it assumes an integral domain embedded into it. $S_n$ is not closed under addition, therefore it is not a ring, what is a prerequisite of being an integral domain.

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  • $\begingroup$ I think that the set of $n$-smooth numbers is only a commutative monoid. To get an abelian group you need the fractions. The answer to your question is that your abelian group includes both the $n$-smooth numbers and fractions of them. $\endgroup$ – Somos May 7 '17 at 21:52
  • $\begingroup$ @Somos thanks for the hint, finally I found a more specific answer that confirms you. $\endgroup$ – kkeri May 14 '17 at 16:02
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I found the answer on the xenharmonic wiki:

For any prime number p, the set of all rational numbers in the p-limit defines a finitely generated free abelian group. The rank of this group is equal to pi(p), the number of prime numbers less than or equal to p.

Prime-limit in music theory is analoguous with the concept of smoothness in number theory.

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