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6.2. HOMOMORPHISMS AND IDEALS
271
In particular, a ring homomorphism is a homomorphism for the abelian
group structure of
R
and
S
, so we know, for example, that
'.
±
x/
D
±
'.x/
and
'.0/
D
0
. Even if
R
and
S
both have an identity element
1
, it is not automatic that
'.1/
D
1
. If we want to specify that this is so,
we will call the homomorphism a
unital
homomorphism.
Example 6.2.2.
The map
'
W
Z
!
Z
n
deﬁned by
'.a/
D
ŒaŁ
D
a
C
n
Z
is a unital ring homomorphism. In fact, it follows from the deﬁnition of
the operations in
Z
n
that
'.a
C
b/
D
Œa
C
bŁ
D
ŒaŁ
C
ŒbŁ
D
'.a/
C
'.b/
,
and, similarly,
'.ab/
D
ŒabŁ
D
ŒaŁŒbŁ
D
'.a/'.b/
for integers
a
and
b
.
Example 6.2.3.
Let
R
be any ring with multiplicative identity
1
. The map
k
7!
k 1
is a ring homomorphism from
Z
to
R
. The map is just the usual
group
homomorphism from
Z
to the additive subgroup
h
1
i
generated by
1
; see Example
2.4.7
. It is necessary to check that
h
1
i
is closed under mul
tiplication and that this map respects multiplication; that is,
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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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