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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
.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
6.2.1. Show that A 7!
and A 7!
are homomorphisms of
the ring of 2-by-2 matrices into the ring of 4-by-4 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 3-by-3 matrices by
a0 a1 a2
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
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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- Fall '08