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
OEaŁ
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
OEa
C
bŁ
D
OEaŁ
C
OEbŁ
D
'.a/
C
'.b/
,
and, similarly,
'.ab/
D
OEabŁ
D
OEaŁOEbŁ
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,
.m 1/.n 1/
D
mn 1
. This follows from two observations:
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 EVERAGE
 Algebra, Abelian group, Identity element, ring homomorphism

Click to edit the document details