College Algebra Exam Review 282

College Algebra Exam Review 282 - 292 6. RINGS Definition...

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Unformatted text preview: 292 6. RINGS Definition 6.4.8. If the subring h1i of an integral domain R generated by the identity is isomorphic to Z, the integral domain is said to have characteristic 0. If the subring h1i is isomorphic to Zp for a prime p , then R is said to have characteristic p . A quotient of an integral domain need not be an integral domain; for example, Z12 is a quotient of Z. On the other hand, a quotient of a ring with nontrivial zero divisors can be an integral domain; Z3 is a quotient of Z12 . Those ideals J in a ring R such that R=J has no nontrivial zero divisors can be easily characterized. An ideal J is said to be prime if for all a; b 2 R, if ab 2 J , then a 2 J or b 2 J . Proposition 6.4.9. Let J be an ideal in a ring R. J is prime if, and only if, R=J has no nontrivial zero divisors. In particular, if R is commutative with identity, then J is prime if, and only if, R=J is an integral domain. Proof. Exercise 6.4.14. I Corollary 6.4.10. In a commutative ring with identity element, every maximal ideal is prime. Proof. If M is a maximal ideal, then R=M is a field by Corollary 6.3.14, and therefore an integral domain. By the proposition, M is a prime ideal. I Example 6.4.11. Every ideal in Z has the form d Z for some d > 0. The ideal d Z is prime if, and only if, d is prime. The proof of this assertion is left as an exercise. Exercises 6.4 6.4.1. Let R be a commutative ring. An element x 2 R is said to be nilpotent if x k D 0 for some natural number k . (a) Show that the set N of nilpotent elements of R is an ideal in R. (b) Show that R=N has no nonzero nilpotent elements. ...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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