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Unformatted text preview: 280 (c)
(d) 6. RINGS Each element x 2 R can be expressed as a sum x D a1 C Cas ,
with ai 2 Ai for all i , in exactly one way.
If 0 D a1 C
C as , with ai 2 Ai for all i , then ai D 0 for all
i. Proof. The equivalence of (a), (c), and (d) is by Proposition 3.5.1 on
page 186. Clearly (b) implies (a). Let us assume (a) and show that the
map
.a1 ; : : : ; as / 7! a1 C C as
is actually a ring isomorphism. We have Ai Aj Â Ai \ Aj D f0g if i ¤ j
(using condition (d)). Therefore,
.a1 C C as /.b1 C C bs / D a1 b1 C C as bs ; whenever ai ; bi 2 Ai for all i . It follows that the map is a ring isomorphism.
I Exercises 6.2
Ä Ä
A0
A0
6.2.1. Show that A 7!
and A 7!
are homomorphisms of
0A
00
the ring of 2by2 matrices into the ring of 4by4 matrices. The former is
unital, but the latter is not.
6.2.2. Deﬁne a map ' from the ring RŒx of polynomials with real coefﬁcients into the ring M of 3by3 matrices by
2
3
a0 a1 a2
X
'.
ai x i / D 4 0 a0 a1 5 :
0 0 a0
Show that ' is a unital ring homomorphism. What is the kernel of this
homomorphism?
6.2.3. If ' W R ! S is a ring homomorphism and R has an identity
element 1, show that e D '.1/ satisﬁes e 2 D e and ex D xe D exe for
all x 2 '.R/.
6.2.4. Show that if ' W R ! S is a ring homomorphism, then '.R/ is a
subring of S .
6.2.5. Show that if ' W R ! S and W S ! T are ring homomorphisms,
then the composition ı ' is a ring homomorphism. ...
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Document
 Fall '08
 EVERAGE
 Algebra

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