h10 - 1 Homework X: August 4, 2009 3.51(i). If A and R are...

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1 Homework X: August 4, 2009 3 . 51 ( i ) . If A and R are domains and ϕ : A R is a ring isomorphism, prove that Φ: [ a, b ] 7→ [ fa, fb ] is a ring isomorphism Frac( A ) Frac( R ). Note first that since f is injective, b 6 = 0 implies fb 6 = 0; hence, [ fa, fb ] makes sense. Next, Φ is well-defined: if [ a, b ] = [ c, d ], then [ fa, fb ] = [ fc, fd ]. This is true: since ad = bc , we have fafd = f ( ad ) = f ( bc ) = fbfc . We check that Φ is a ring map. Φ[1 , 1] = [ f 1 , f 1] = [1 , 1]; Φ([ a, b ] + [ c, d ]) = Φ[ ad + bc, bd ] = [ f ( ad + bc ) , f ( bd )] = [ fafd + fbfc, fbfd ] = [ fa, fb ] + [ fc, fd ] = Φ[ a, b ] + Φ[ c, d ]. Similarly, Φ preserves multiplication. To see that Φ is an isomorphism, we exhibit its inverse Φ - 1 . Define Ψ: Frac( R ) Frac( A ) by [ r, s ] 7→ [ f - 1 r, f - 1 s ]. ( ii ) . Show that a field k containing an isomorphic copy of Z as a subring must contain an isomorphic copy of Q . By hypothesis, there is an isomorphism f : Z R , where R is a subring of k . It follows from (i) that there is an isomorphism f : Frac( Z ) R 0 , where R 0 is a subfield of k containing im f . But Frac(
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This note was uploaded on 11/15/2010 for the course MATH 417 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.

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