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faradayslaw
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Homework Statement
Show that the ring of rational numbers whose reduced form denominator is not divisble by a prime, p, mod an ideal the set of elements of the above set whose numerators are divisible by p is isomorphic to Z_p
Homework Equations
The Attempt at a Solution
It seems very trivial: Use 1st homomorphism theorem with phi(a/b) = a(modp), but I am having a hard time showing that such a mapping is actually a homomorphism additively. I.E., phi(a/b + c/d) = phi(ad+bc/bd) = ad+bc mod(p) =/= a modp + b modp = phi(a/b) + phi (c/d).
I am stuck here and any help would be appreciated.
Thanks,