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October 18th, 2011, 09:29 AM   #1
Joined: Oct 2009

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homomorphism, fraction ring


I would really apreciate some help with this...

f is a ring homomorphism between two commutative rings A and B. So f: A --->B
S is a multiplicatively closed subset of A. Define T:= f(S) then T is a multiplicatively closed subset of B.

The book says that there is a homomorphism ,say h, between the ring of fractions S^(-1)B and T^(-1)B which sends b/s to b/f(s). In fact this is an isomorphism. And S^(-1)B and T^(-1)B are isomorphic as S^(-1)A-modules.

I already get stuck at this:

h((b1/s1) + (b2/s2))= h ((b1s2 + b2s1)/s1s2)= (b1s2 + b2s1)/f(s1s2) = (b1s2)/(f(s1)f(s2)) + (b2s1)/(f(s1)f(s2)) how is this equal to h(b1/s1) + h(b2/s2)

can I do this: (b1s2)/(f(s1)f(s2)) = (b1/f(s1)) s2(1/f(s2))= (b1/f(s1)) s2(f(1)/f(s2)) = (b1/f(s1)) (f(s2)/f(s2)) ???
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fraction, homomorphism, ring

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