This preview shows page 1. Sign up to view the full content.
Unformatted text preview: 6.2. HOMOMORPHISMS AND IDEALS 275 is defined similarly. Note that for commutative rings, all of these notions coincide. Proposition 6.2.13. If ' W R ! S is a ring homomorphism, then ker .'/ is an ideal of of R . Proof. Since ' is a homomorphism of abelian groups, its kernel is a sub- group. If r 2 R and x 2 ker .'/ , then '.rx/ D '.r/'.x/ D '.r/0 D . Hence rx 2 ker .'/ . Similarly, , xr 2 ker .'/ . n Example 6.2.14. The kernel of the ring homomorphism Z ! Z n given by k 7! OEkŁ is n Z . Example 6.2.15. Let R be any ring with multiplicative identity element. Consider the unital ring homomorphism from Z to R defined by k 7! k 1 . Note that if k 1 D , then for all a 2 R , k a D .k 1/a D 0a D , by the “elementary deductions” on pages 264 – 265 . Therefore the kernel coincides with f k 2 Z W k a D for all a 2 R g Since the kernel is a subgroup of Z , it is equal to n Z for a unique n , according to Proposition 2.2.21 on page 98 . The integer n is called the characteristic of R . The characteristic is....
View Full Document
This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
- Fall '08