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Unformatted text preview: Solution. Let A B C be ring homomorphisms. Now : 1 7 ( 1 ) = 1 7 ( 1 ) = 1 , so that the composite preserves 1. Let a , a A . Then : a + a 7 ( a + a ) = ( a ) + ( a ) 7 (v ar phi ( a ) + ( a )) = (v ar phi ( a )) + (( a )). One shows that the composite preserves multiplication in the same way, and one concludes that the composite of ring homomorphisms is again a ring homomorphism. Since the composite of bijections is always a bijection, it follows that the composite of ring isomor-phisms is again an isomorphism. (iii) Show that A = R de f nes an equivalence relation on any family of commutative rings. Solution. (i) Re F exive: If A is a ring, then the identity map 1 A : A A is an isomorphism....
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.
- Fall '11