7/23/2019 Cuarta Tarea Anillos y Campos
1/3
Sec. 3.5 Homomorphisms 149
Definition. If f and g are elements in a commutative ring R,
then a common multiple
is an element m R with f | m and g | m. If f and g in R are not
both 0, define their
least common multiple, abbreviated lcm, to be a common multiple
c of them with c | mfor every common multiplem . If f =0 = g ,
define their lcm = 0. The lcm of f andg is
often denoted by [f, g].
Least common multiples, when they exist, need not be unique; for
example, it is easy
to see that ifc is a least common multiple of f andg , then so
is ucfor any unit u R. In
the special case R = Z, we force uniqueness of the lcm by
requiring it to be positive; if
R = k[x], wherekis a field, then we force uniqueness of the lcm
by further requiring it to
be monic.
EXERCISES
3.39 (i) Let : A R be an isomorphism, and let : R A be its
inverse. Show that is
an isomorphism.(ii) Show that the composite of two homomorphisms
(isomorphisms) is again a homomor-
phism (isomorphism).
(iii) Show thatA = R defines an equivalence relation on the
class of all commutative rings.
3.40 Let R be a commutative ring and let F(R)be the commutative
ring of all functions f: R R
with pointwise operations.
(i) Show thatR is isomorphic to the subring ofF(R)consisting of
all the constant func-
tions.
(ii) If f(x) R[x], let f: R R be defined byr f(r)[thus, fis the
polynomial
function associated to f(x)]. Show that the function : R[x]
F(R), defined by
( f(x))= f, is a ring homomorphism.
(iii) Show that is injective ifR is an infinite field.
3.41 Let I and Jbe nonzero ideals in a commutative ring R . IfR
is a domain, prove that I J =
{0}.3.42 Let R be a commutative ring. Show that the function :
R[x] R, defined by
: a0+ a1x + a2x + + anxn a0,
is a homomorphism. Describe ker in terms of roots of
polynomials.
3.43 IfR is a commutative ring and c R, prove that the function
: R[x] R[x], defined by
f(x) f(x + c), is an isomorphism. In more detail,(
i sixi )=
i si (x + c)i .
Hint. This is a routine but long calculation.
3.44 (i) Prove that F, the field with four elements (see
Exercise 3.14 on page 125), and I4 are
not isomorphic commutative rings.
(ii) Prove that any twofields having exactly four elements are
isomorphic.
Hint. First prove that 1 + 1= 0, and then show that the nonzero
elements form a cyclic
group of order 3 under multiplication.
3.45 (i) Show that every elementa Ip has a pth root (i.e., there
is b Ip witha = bp).(ii) Letkbe a field that contains Ip as a
subfield [e.g.,k= Ip(x)]. For every positive integer
n, show that the functionn: k k, given by(a)= apn , is a ring
homomorphism.
Reyn
Aless
Migue
Rocio
Glori
Reyn
Aless
7/23/2019 Cuarta Tarea Anillos y Campos
3/3
Sec. 3.6 Euclidean Rings 151
3.53 IfR is a PID and a, b R, prove that their lcm exists.
3.54 (i) Ifkis a field, prove that the ring of formal power
series k[[x]] is a PID.
Hint. If I is a nonzero ideal, choose Iof smallest order. Use
Exercise 3.27 onpage 130 to prove that I =( ).
(ii) Prove that every nonzero ideal ink[[x]] is equal to (xn)for
somen 0.
3.55 Ifkis a field, show that the ideal(x,y)in k[x,y] is not a
principal ideal (see page 129).
3.56 For everym 1, prove that every ideal in Im is a principal
ideal. (Ifm is composite, then Imis not a PID because it is not a
domain.)
3.6 EUCLIDEAN R INGS
There are rings other than Z andk[x], wherekis a field, that
have a division algorithm. In
particular, we present an example of such a ring in which the
quotient and remainder are
not unique. We begin by generalizing a property shared by both Z
andk[x].
Definition. Aeuclidean ringis a domain R that is equipped with a
function
: R {0} N,
called adegree function, such that
(i) (f) (f g)for all f, g Rwith f, g =0;
(ii) for all f, g R with f =0, there existq ,r Rwith
g = q f + r,
where eitherr =0 or (r) < (f).
Note that ifR has a degree function that is identically 0, then
condition (ii) forces
r =0 always; taking g = 1 shows that Ris a field in this
case.
Example 3.59.
(i) The integers Z is a euclidean ring with degree function (m)=
|m|. In Z, we have
(mn)= |mn | = |m||n| = (m)(n).
(ii) When k is a field, the domain k[x] is a euclidean ring with
degree function the usual
degree of a nonzero polynomial. In k[x], we have
( f g)= deg(f g)
=deg(f) + deg(g)
= (f) + (g).
Glori
Reyn
Aless
Migue
