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College Algebra Exam Review 261

College Algebra Exam Review 261 - 6.2 HOMOMORPHISMS AND...

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6.2. HOMOMORPHISMS AND IDEALS 271 In particular, a ring homomorphism is a homomorphism for the abelian group structure of R and S , so we know, for example, that '. x/ D '.x/ and '.0/ D 0 . Even if R and S both have an identity element 1 , it is not automatic that '.1/ D 1 . If we want to specify that this is so, we will call the homomorphism a unital homomorphism. Example 6.2.2. The map ' W Z ! Z n defined by '.a/ D OEaŁ D a C n Z is a unital ring homomorphism. In fact, it follows from the definition of the operations in Z n that '.a C b/ D OEa C D OEaŁ C OEbŁ D '.a/ C '.b/ , and, similarly, '.ab/ D OEabŁ D OEaŁOEbŁ D '.a/'.b/ for integers a and b . Example 6.2.3. Let R be any ring with multiplicative identity 1 . The map k 7! k 1 is a ring homomorphism from Z to R . The map is just the usual group homomorphism from Z to the additive subgroup h 1 i generated by 1 ; see Example 2.4.7 . It is necessary to check that h 1 i is closed under mul- tiplication and that this map respects multiplication; that is, .m 1/.n 1/ D mn 1 . This follows from two observations:
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