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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.
 Fall '08
 EVERAGE
 Algebra

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