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Unformatted text preview: 294 6.4.12.
(a)
(b) 6. RINGS Show that a 7! Œa=1 is an injective unital ring homomorphism
of R into Q.R/. In this sense Q.R/ is a ﬁeld containing R.
If F is a ﬁeld such that F Ã R, show that there is an injective
unital homomorphism ' W Q.R/ ! F such that '.Œa=1/ D a
for a 2 R. 6.4.13. If R is the ring of Gaussian integers, show that Q.R/ is isomorphic
to the subﬁeld of C consisting of complex numbers with rational real and
imaginary parts.
6.4.14. Let J be an ideal in a ring R. Show that 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.)
6.4.15. Every ideal in Z has the form d Z for some d > 0. Show that the
ideal d Z is prime if, and only if, d is prime.
6.4.16. Show that a maximal ideal in a commutative ring with identity is
prime. 6.5. Euclidean Domains, Principal Ideal
Domains, and Unique Factorization
We have seen two examples of integral domains with a good theory of
factorization, the ring of integers Z and the ring of polynomials KŒx over
a ﬁeld K . In both of these rings R, every nonzero, noninvertible element
has an essentially unique factorization into irreducible factors.
The common feature of these rings, which was used to establish unique
factorization is a Euclidean function d W R n f0g ! N [ f0g with the
property that d.fg/ maxfd.f /; d.g/g and for each f; g 2 R n f0g there
exist q; r 2 R such that f D qg C r and r D 0 or d.r/ < d.g/.
For the integers, the Euclidean function is d.n/ D jnj. For the polynomials, the function is d.f / D deg.f /.
Deﬁnition 6.5.1. Call an integral domain R a Euclidean domain if it admits a Euclidean function.
Let us consider one more example of a Euclidean domain, in order to
justify having made the deﬁnition.
Example 6.5.2. Let ZŒi be the ring of Gaussian integers, namely, the
complex numbers with integer real and imaginary parts. For the ”degree” map d take d.z/ D jz j2 . We have d.zw/ D d.z/d.w/. The ...
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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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