College Algebra Exam Review 281

College Algebra Exam Review 281 - 6.4. INTEGRAL DOMAINS 291...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 6.4. INTEGRAL DOMAINS 291 any injective homomorphism of R into a field F extends to an injective homomorphism of Q.R/ into F : ' F R qqqqq qqqq qqq qqqqq qqqqqqqqqq qqq q  ' Q qqqqq qqqqq qqqqqqqq Q.R/ The main steps in establishing these facts are the following: 1. Q.R/ is a ring with zero element 0 D Œ0=1 and multiplicative identity 1 D Œ1=1. In Q.R/, 0 ¤ 1. 2. Œa=b D 0 if, and only if, a D 0. If Œa=b ¤ 0, then Œa=bŒb=a D 1. Thus Q.R/ is a field. 3. a 7! Œa=1 is an injective homomorphism of R into Q.R/. 4. If ' W R ! F is an injective homomorphism of R into a field F , then ' W Œa=b 7! '.a/='.b/ defines an injective homomorphism Q of Q.R/ into F , which extends ' . See Exercises 6.4.11 and 6.4.12. Proposition 6.4.4. If R is an integral domain, then Q.R/ is a field containing R as a subring. Moreover, any injective homomorphism of R into a field F extends to an injective homomorphism of Q.R/ into F . Example 6.4.5. Q.Z/ D Q. Example 6.4.6. Q.KŒx/ is the field of rational functions in one variable. This field is denoted K.x/. Example 6.4.7. Q.KŒx1 ; : : : ; xn / is the field of rational functions in n variables. This field is denoted K.x1 ; : : : ; xn /. We have observed in Example 6.2.3 that in a ring R with multiplicative identity 1, the additive subgroup h1i generated by 1 is a subring and the map k 7! k 1 is a ring homomorphism from Z onto h1i  R. If this ring homomorphism is injective, then h1i Š Z as rings. Otherwise, the kernel of the homomorphism is nZ for some n 2 N and h1i Š Z=nZ D Zn as rings. If R is an integral domain, then any subring containing the identity is also an integral domain, so, in particular, h1i is an integral domain. If h1i is finite, so ring isomorphic to Zn for some n, then n must be prime. (Zn is an integral domain if, and only if, n is prime.) ...
View Full Document

Ask a homework question - tutors are online